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Type | Label | Description | ||||||||||||||||||||||
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Statement | ||||||||||||||||||||||||
Theorem | csbfinxpg 32401* | Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁)) | ||||||||||||||||||||||||
Theorem | finxpreclem1 32402* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_{𝑜} ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^{st} ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_{𝑜}, 𝑋⟩)) | ||||||||||||||||||||||||
Theorem | finxpreclem2 32403* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_{𝑜} ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^{st} ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1_{𝑜}, 𝑋⟩)) | ||||||||||||||||||||||||
Theorem | finxp0 32404 | The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑∅) = ∅ | ||||||||||||||||||||||||
Theorem | finxp1o 32405 | The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑1_{𝑜}) = 𝑈 | ||||||||||||||||||||||||
Theorem | finxpreclem3 32406* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_{𝑜} ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^{st} ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) ⇒ ⊢ (((𝑁 ∈ ω ∧ 2_{𝑜} ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1^{st} ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩)) | ||||||||||||||||||||||||
Theorem | finxpreclem4 32407* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_{𝑜} ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^{st} ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) ⇒ ⊢ (((𝑁 ∈ ω ∧ 2_{𝑜} ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1^{st} ‘𝑦)⟩)‘∪ 𝑁)) | ||||||||||||||||||||||||
Theorem | finxpreclem5 32408* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_{𝑜} ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^{st} ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) ⇒ ⊢ ((𝑛 ∈ ω ∧ 1_{𝑜} ∈ 𝑛) → (¬ 𝑥 ∈ (V × 𝑈) → (𝐹‘⟨𝑛, 𝑥⟩) = ⟨𝑛, 𝑥⟩)) | ||||||||||||||||||||||||
Theorem | finxpreclem6 32409* | Lemma for ↑↑ recursion theorems. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_{𝑜} ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^{st} ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) ⇒ ⊢ ((𝑁 ∈ ω ∧ 1_{𝑜} ∈ 𝑁) → (𝑈↑↑𝑁) ⊆ (V × 𝑈)) | ||||||||||||||||||||||||
Theorem | finxpsuclem 32410* | Lemma for finxpsuc 32411. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1_{𝑜} ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1^{st} ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) ⇒ ⊢ ((𝑁 ∈ ω ∧ 1_{𝑜} ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) | ||||||||||||||||||||||||
Theorem | finxpsuc 32411 | The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) | ||||||||||||||||||||||||
Theorem | finxp2o 32412 | The value of Cartesian exponentiation at two. (Contributed by ML, 19-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑2_{𝑜}) = (𝑈 × 𝑈) | ||||||||||||||||||||||||
Theorem | finxp3o 32413 | The value of Cartesian exponentiation at three. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (𝑈↑↑3_{𝑜}) = ((𝑈 × 𝑈) × 𝑈) | ||||||||||||||||||||||||
Theorem | finxpnom 32414 | Cartesian exponentiation when the exponent is not a natural number defaults to the empty set. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (¬ 𝑁 ∈ ω → (𝑈↑↑𝑁) = ∅) | ||||||||||||||||||||||||
Theorem | finxp00 32415 | Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.) | ||||||||||||||||||||||
⊢ (∅↑↑𝑁) = ∅ | ||||||||||||||||||||||||
Theorem | wl-section-prop 32416 |
Intuitionistic logic is now developed separately, so we need not first
focus on intuitionally valid axioms ax-1 6 and
ax-2 7
any longer.
Alternatively, I start from Jan Lukasiewicz's axiom system here, i.e. ax-mp 5, ax-luk1 32417, ax-luk2 32418 and ax-luk3 32419. I rather copy this system than use luk-1 1571 to luk-3 1573, since the latter are theorems, while we need axioms here. (Contributed by Wolf Lammen, 23-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||||||||||||||||||||||||
Axiom | ax-luk1 32417 |
1 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-1 1571 and imim1 81, but introduced as an axiom. It
focuses on a basic
property of a valid implication, namely that the consequent has to be true
whenever the antecedent is. So if 𝜑 and 𝜓 are somehow
parametrized expressions, then 𝜑 → 𝜓 states that 𝜑 strengthen
𝜓, in that 𝜑 holds only for a (often
proper) subset of those
parameters making 𝜓 true. We easily accept, that when
𝜓 is
stronger than 𝜒 and, at the same time 𝜑 is
stronger than
𝜓, then 𝜑 must be stronger than
𝜒.
This transitivity is
expressed in this axiom.
A particular result of this strengthening property comes into play if the antecedent holds unconditionally. Then the consequent must hold unconditionally as well. This specialization is the foundational idea behind logical conclusion. Such conclusion is best expressed in so-called immediate versions of this axiom like imim1i 61 or syl 17. Note that these forms are weaker replacements (i.e. just frequent specialization) of the closed form presented here, hence a mere convenience. We can identify in this axiom up to three antecedents, followed by a consequent. The number of antecedents is not really fixed; the fewer we are willing to "see", the more complex the consequent grows. On the other side, since 𝜒 is a variable capable of assuming an implication itself, we might find even more antecedents after some substitution of 𝜒. This shows that the ideas of antecedent and consequent in expressions like this depends on, and can adapt to, our current interpretation of the whole expression. In this axiom, up to two antecedents happen to be of complex nature themselves, i.e. are an embedded implication. Logically, this axiom is a compact notion of simpler expressions, which I will later coin implication chains. Herein all antecedents and the consequent appear as simple variables, or their negation. Any propositional expression is equivalent to a set of such chains. This axiom, for example, is dissected into following chains, from which it can be recovered losslessly: (𝜓 → (𝜒 → (𝜑 → 𝜒))); (¬ 𝜑 → (𝜒 → (𝜑 → 𝜒))); (𝜓 → (¬ 𝜓 → (𝜑 → 𝜒))); (¬ 𝜑 → (¬ 𝜓 → (𝜑 → 𝜒))). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||||||||||||||||||||||||
Axiom | ax-luk2 32418 |
2 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-2 1572 or pm2.18 121, but introduced as an axiom. The core idea
behind
this axiom is, that if something can be implied from both an antecedent,
and separately from its negation, then the antecedent is irrelevant to the
consequent, and can safely be dropped. This is perhaps better seen from
the following slightly extended version (related to pm2.65 183):
((𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)). (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||||||||||||||||||||||||
Axiom | ax-luk3 32419 |
3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of
luk-3 1573 and pm2.24 120, but introduced as an axiom.
One might think that the similar pm2.21 119 (¬ 𝜑 → (𝜑 → 𝜓)) is
a valid replacement for this axiom. But this is not true, ax-3 8 is not
derivable from this modification.
This can be shown by designing carefully operators ¬ and → on a
finite set of primitive statements. In propositional logic such
statements are ⊤ and ⊥, but we can assume more and other
primitives in our universe of statements. So we denote our primitive
statements as phi0 , phi1 and phi2. The actual meaning of the statements
are not important in this context, it rather counts how they behave under
our operations ¬ and →, and which of them we assume to hold
unconditionally (phi1, phi2). For our disproving model, I give that
information in tabular form below. The interested reader may check per
hand, that all possible interpretations of ax-mp 5, ax-luk1 32417, ax-luk2 32418
and pm2.21 119 result in phi1 or phi2, meaning they always hold. But for
wl-ax3 32431 we can find a counter example resulting in phi0, not a statement
always true.
The verification of a particular set of axioms in a given model is tedious
and error prone, so I wrote a computer program, first checking this for
me,
and second, hunting for a counter example. Here is the result, after 9165
fruitlessly computer generated models:
ax-3 fails for phi2, phi2 number of statements: 3 always true phi1 phi2 Negation is defined as ----------------------------------------------------------------------
Implication is defined as ----------------------------------------------------------------------
(Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-section-boot 32420 | In this section, I provide the first steps needed for convenient proving. The presented theorems follow no common concept other than being useful in themselves, and apt to rederive ax-1 6, ax-2 7 and ax-3 8. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||||||||||||||||||||||||
Theorem | wl-imim1i 32421 | Inference adding common consequents in an implication, thereby interchanging the original antecedent and consequent. Copy of imim1i 61 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 → 𝜒) → (𝜑 → 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-syl 32422 | An inference version of the transitive laws for implication luk-1 1571. Copy of syl 17 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-syl5 32423 | A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. Copy of syl5 33 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → (𝜓 → 𝜃)) ⇒ ⊢ (𝜒 → (𝜑 → 𝜃)) | ||||||||||||||||||||||||
Theorem | wl-pm2.18d 32424 | Deduction based on reductio ad absurdum. Copy of pm2.18d 123 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (¬ 𝜓 → 𝜓)) ⇒ ⊢ (𝜑 → 𝜓) | ||||||||||||||||||||||||
Theorem | wl-con4i 32425 | Inference rule. Copy of con4i 112 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-pm2.24i 32426 | Inference rule. Copy of pm2.24i 145 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ (¬ 𝜑 → 𝜓) | ||||||||||||||||||||||||
Theorem | wl-a1i 32427 | Inference rule. Copy of a1i 11 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-mpi 32428 | A nested modus ponens inference. Copy of mpi 20 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜓 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-imim2i 32429 | Inference adding common antecedents in an implication. Copy of imim2i 16 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 → 𝜑) → (𝜒 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-syl6 32430 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. Copy of syl6 34 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 → 𝜃)) | ||||||||||||||||||||||||
Theorem | wl-ax3 32431 | ax-3 8 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-ax1 32432 | ax-1 6 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-pm2.27 32433 | This theorem, called "Assertion," can be thought of as closed form of modus ponens ax-mp 5. Theorem *2.27 of [WhiteheadRussell] p. 104. Copy of pm2.27 41 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-com12 32434 | Inference that swaps (commutes) antecedents in an implication. Copy of com12 32 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜓 → (𝜑 → 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-pm2.21 32435 | From a wff and its negation, anything follows. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. Copy of pm2.21 119 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-con1i 32436 | A contraposition inference. Copy of con1i 143 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → 𝜓) ⇒ ⊢ (¬ 𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-ja 32437 | Inference joining the antecedents of two premises. Copy of ja 172 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 → 𝜒) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-imim2 32438 | A closed form of syllogism (see syl 17). Theorem *2.05 of [WhiteheadRussell] p. 100. Copy of imim2 56 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) | ||||||||||||||||||||||||
Theorem | wl-a1d 32439 | Deduction introducing an embedded antecedent. Copy of imim2 56 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-ax2 32440 | ax-2 7 proved from Lukasiewicz's axioms. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-id 32441 | Principle of identity. Theorem *2.08 of [WhiteheadRussell] p. 101. Copy of id 22 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-notnotr 32442 | Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. In classical logic (our logic) this is always true. In intuitionistic logic this is not always true; in intuitionistic logic, when this is true for some 𝜑, then 𝜑 is stable. Copy of notnotr 124 with a different proof. (Contributed by Wolf Lammen, 17-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (¬ ¬ 𝜑 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-pm2.04 32443 | Swap antecedents. Theorem *2.04 of [WhiteheadRussell] p. 100. This was the third axiom in Frege's logic system, specifically Proposition 8 of [Frege1879] p. 35. Copy of pm2.04 88 with a different proof. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-section-impchain 32444 |
An implication like (𝜓 → 𝜑) with one antecedent can easily be
extended by prepending more and more antecedents, as in
(𝜒
→ (𝜓 → 𝜑)) or (𝜃 → (𝜒 → (𝜓 → 𝜑))). I
call these expressions implication chains, and the number of antecedents
(number of nodes minus one) denotes their length. A given length often
marks just a required minimum value, since the consequent 𝜑 itself
may represent an implication, or even an implication chain, such hiding
part of the whole chain. As an extension, it is useful to consider a
single variable 𝜑 as a degenerate implication chain of
length zero.
Implication chains play a particular role in logic, as all propositional expressions turn out to be convertible to one or more implication chains, their nodes as simple as a variable, or its negation. So there is good reason to focus on implication chains as a sort of normalized expressions, and build some general theorems around them, with proofs using recursive patterns. This allows for theorems referring to longer and longer implication chains in an automated way. The theorem names in this section contain the text fragment 'impchain' to point out their relevance to implication chains, followed by a number indicating the (minimal) length of the longest chain involved. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-x 32445 | This series of theorems provide a means of exchanging the consequent of an implication chain via a simple implication. In the main part, the theorems ax-mp 5, syl 17, syl6 34, syl8 74 form the beginning of this series. These theorems are replicated here, but with proofs that aim at a recursive scheme, allowing to base a proof on that of the previous one in the series. (Contributed by Wolf Lammen, 17-Nov-2019.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-0 32446 |
This theorem is the start of a proof recursion scheme where we replace
the consequent of an implication chain. The number '0' in the theorem
name indicates that the modified chain has no antecedents.
This theorem is in fact a copy of ax-mp 5, and is repeated here to emphasize the recursion using similar theorem names. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜓 & ⊢ (𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-1 32447 | This theorem is in fact a copy of wl-syl 32422, and repeated here to demonstrate a recursive proof scheme. The number '1' in the theorem name indicates that a chain of length 1 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜒 → 𝜓) & ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜒 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-impchain-mp-2 32448 | This theorem is in fact a copy of wl-syl6 32430, and repeated here to demonstrate a recursive proof scheme. The number '2' in the theorem name indicates that a chain of length 2 is modified. (Contributed by Wolf Lammen, 6-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜃 → (𝜒 → 𝜓)) & ⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜃 → (𝜒 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.x 32449 |
It is often convenient to have the antecedent under focus in first
position, so we can apply immediate theorem forms (as opposed to
deduction, tautology form). This series of theorems swaps the first with
the last antecedent in an implication chain. This kind of swapping is
self-inverse, whence we prefer it over, say, rotating theorems. A
consequent can hide a tail of a longer chain, so theorems of this series
appear as swapping a pair of antecedents with fixed offsets. This form of
swapping antecedents is flexible enough to allow for any permutation of
antecedents in an implication chain.
The first elements of this series correspond to com12 32, com13 86, com14 94 and com15 99 in the main part. The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-mp-x 32445 series developed before. (Contributed by Wolf Lammen, 17-Nov-2019.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.1 32450 |
A degenerate form of antecedent swapping. The number '1' in the theorem
name indicates that it handles a chain of length 1.
Since there is just one antecedent in the chain, there is nothing to swap. Non-degenerated forms begin with wl-impchain-com-1.2 32451, for more see there. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜓 → 𝜑) ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.2 32451 |
This theorem is in fact a copy of wl-com12 32434, and repeated here to
demonstrate a simple proof scheme. The number '2' in the theorem name
indicates that a chain of length 2 is modified.
See wl-impchain-com-1.x 32449 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜒 → (𝜓 → 𝜑)) ⇒ ⊢ (𝜓 → (𝜒 → 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.3 32452 |
This theorem is in fact a copy of com13 86, and repeated here to
demonstrate a simple proof scheme. The number '3' in the theorem name
indicates that a chain of length 3 is modified.
See wl-impchain-com-1.x 32449 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜓 → (𝜒 → (𝜃 → 𝜑))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-1.4 32453 |
This theorem is in fact a copy of com14 94, and repeated here to
demonstrate a simple proof scheme. The number '4' in the theorem name
indicates that a chain of length 4 is modified.
See wl-impchain-com-1.x 32449 for more information how this proof is generated. (Contributed by Wolf Lammen, 7-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜂 → (𝜃 → (𝜒 → (𝜓 → 𝜑)))) ⇒ ⊢ (𝜓 → (𝜃 → (𝜒 → (𝜂 → 𝜑)))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-n.m 32454 |
This series of theorems allow swapping any two antecedents in an
implication chain. The theorem names follow a pattern wl-impchain-com-n.m
with integral numbers n < m, that swaps the m-th antecedent with n-th
one
in an implication chain. It is sufficient to restrict the length of the
chain to m, too, since the consequent can be assumed to be the tail right
of the m-th antecedent of any arbitrary sized implication chain. We
further assume n > 1, since the wl-impchain-com-1.x 32449 series already
covers the special case n = 1.
Being able to swap any two antecedents in an implication chain lays the foundation of permuting its antecedents arbitrarily. The proofs of this series aim at automated proofing using a simple scheme. Any instance of this series is a triple step of swapping the first and n-th antecedent, then the first and the m-th, then the first and the n-th antecedent again. Each of these steps is an instance of the wl-impchain-com-1.x 32449 series. (Contributed by Wolf Lammen, 17-Nov-2019.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-com-2.3 32455 | This theorem is in fact a copy of com23 84. It starts a series of theorems named after wl-impchain-com-n.m 32454. For more information see there. (Contributed by Wolf Lammen, 12-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜃 → (𝜓 → (𝜒 → 𝜑))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-2.4 32456 | This theorem is in fact a copy of com24 93. It is another instantiation of theorems named after wl-impchain-com-n.m 32454. For more information see there. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜂 → (𝜃 → (𝜒 → (𝜓 → 𝜑)))) ⇒ ⊢ (𝜂 → (𝜓 → (𝜒 → (𝜃 → 𝜑)))) | ||||||||||||||||||||||||
Theorem | wl-impchain-com-3.2.1 32457 | This theorem is in fact a copy of com3r 85. The proof is an example of how to arrive at arbitrary permutations of antecedents, using only swapping theorems. The recursion principle is to first swap the correct antecedent to the position just before the consequent, and then employ a theorem handling an implication chain of length one less to reorder the others. (Contributed by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜃 → (𝜒 → (𝜓 → 𝜑))) ⇒ ⊢ (𝜓 → (𝜃 → (𝜒 → 𝜑))) | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-x 32458 |
If an implication chain is assumed (hypothesis) or proven (theorem) to
hold, then we may add any extra antecedent to it, without changing its
truth. This is expressed in its simplest form in wl-a1i 32427, that allows
us prepending an arbitrary antecedent to an implication chain. Using our
antecedent swapping theorems described in wl-impchain-com-n.m 32454, we may
then move such a prepended antecedent to any desired location within all
antecedents. The first series of theorems of this kind adds a single
antecedent somewhere to an implication chain. The appended number in the
theorem name indicates its position within all antecedents, 1 denoting the
head position. A second theorem series extends this idea to multiple
additions (TODO).
Adding antecedents to an implication chain usually weakens their universality. The consequent afterwards dependends on more conditions than before, which renders the implication chain less versatile. So you find this proof technique mostly when you adjust a chain to a hypothesis of a rule. A common case are syllogisms merging two implication chains into one. The first elements of the first series correspond to a1i 11, a1d 25 and a1dd 48 in the main part. The proofs of this series aim at automated proving using a simple recursive scheme. It employs the previous theorem in the series along with a sample from the wl-impchain-com-1.x 32449 series developed before. (Contributed by Wolf Lammen, 20-Jun-2020.) | ||||||||||||||||||||||
⊢ ⊤ | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-1 32459 | Inference rule, a copy of a1i 11. Head start of a recursive proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ (𝜓 → 𝜑) | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-2 32460 | Inference rule, a copy of a1d 25. First recursive proof based on the previous instance. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 → 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-impchain-a1-3 32461 | Inference rule, a copy of a1dd 48. A recursive proof depending on previous instances, and demonstrating the proof pattern. (Contributed by Wolf Lammen, 20-Jun-2020.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-section-nf 32462 |
The current definition of 'not free', nf5 2102, has its downsides. In
particular, it often drags axioms ax-10 2006 and ax-12 2034 into proofs, that
are not needed otherwise. This is because of the particular structure
of the term ∀𝑥(𝜑 → ∀𝑥𝜑). It does not allow an easy
transition 𝜑--> ¬ 𝜑 (see nfn 1768).
The mix of both
quantified and simple 𝜑 requires explicit use of sp 2041 (or
ax-12 2034)
in many instances. And, finally, the nesting of the quantifier ∀𝑥
sometimes requires an invocation of theorems like nfa1 2015.
All of this
mandates the use of ax-10 2006 and/or ax-12 2034.
On the other hand, the current definition is structurally better aligned with both the hb* series of theorems and ax-5 1827. A note on ax-10 2006: The obvious content of this axiom is, that the ¬ operator does not change the not-free state of a set varable. This is in fact only one aspect of this axiom. The other one, more hidden, states that 𝑥 is not free in ∀𝑥𝜑. A simple transformation renders this axiom as (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑), from which the second aspect is better deduced. Both aspects are not needed in simple applications of the df-nf 1701 style definition, while use of the current nf5 2102 incurs these. A note on ax-12 2034: This axiom enters proofs using Ⅎ via sp 2041 or 19.8a 2039 or their variants. In a context where both mixed quantified and simple variables 𝜑 appear (like 19.21 2062), this axiom is almost always required, no matter how 'not free' is defined. But in a context, where a variable 𝜑 appears only quantified, chances are, this axiom can be evaded when using df-nf 1701, but not when using nf5 2102. (Contributed by Wolf Lammen, 11-Sep-2021.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||||||||||||||||||||||||
Theorem | wl-nf-nf2 32463 | By ax-10 2006 the definition nf5 2102 is stricter than the df-nf 1701 style. (Contributed by Wolf Lammen, 14-Sep-2021.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) ⇒ ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nf2-nf 32464 | hba1 2137 is sufficient to let a df-nf 1701 style definition be stricter than nf5 2102. (Contributed by Wolf Lammen, 14-Sep-2021.) | ||||||||||||||||||||||
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) ⇒ ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑)) | ||||||||||||||||||||||||
Axiom | ax-wl-13v 32465* |
A version of ax13v 2235 with a distinctor instead of a distinct
variable
expression.
Had we additonally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1827. So essentially this theorem states, that a distinct variable condition between set variables can be replaced with a distinctor expression. (Contributed by Wolf Lammen, 23-Jul-2021.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||||||||||||||||||||||||
Theorem | wl-ax13lem1 32466* | A version of ax-wl-13v 32465 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-jarri 32467 | Dropping a nested antecedent. This theorem is one of two reversions of ja 172. Since ja 172 is reversible, a nested (chain of) implication(s) is just a packed notation of two or more theorems/hypotheses with a common consequent. axc5c7 33214 is an instance of this idea. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → 𝜓) → 𝜒) ⇒ ⊢ (𝜓 → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-jarli 32468 | Dropping a nested consequent. This theorem is one of two reversions of ja 172. Since ja 172 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. axc5c7 33214 is an instance of this idea. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ ((𝜑 → 𝜓) → 𝜒) ⇒ ⊢ (¬ 𝜑 → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-mps 32469 | Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜃) | ||||||||||||||||||||||||
Theorem | wl-syls1 32470 | Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜓 → 𝜒) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜃) | ||||||||||||||||||||||||
Theorem | wl-syls2 32471 | Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ (𝜑 → 𝜓) & ⊢ ((𝜑 → 𝜒) → 𝜃) ⇒ ⊢ ((𝜓 → 𝜒) → 𝜃) | ||||||||||||||||||||||||
Theorem | wl-embant 32472 | A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||
⊢ 𝜑 & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ((𝜑 → 𝜓) → 𝜒) | ||||||||||||||||||||||||
Theorem | wl-orel12 32473 | In a conjunctive normal form a pair of nodes like (𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒) eliminates the need of a node (𝜓 ∨ 𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.) | ||||||||||||||||||||||
⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒)) | ||||||||||||||||||||||||
Theorem | wl-cases2-dnf 32474 | A particular instance of orddi 909 and anddi 910 converting between disjunctive and conjunctive normal forms, when both 𝜑 and ¬ 𝜑 appear. This theorem in fact rephrases cases2 1005, and is related to consensus 990. I restate it here in DNF and CNF. The proof deliberately does not use df-ifp 1007 and dfifp4 1010, by which it can be shortened. (Contributed by Wolf Lammen, 21-Jun-2020.) (Proof modification is discouraged.) | ||||||||||||||||||||||
⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-dfnan2 32475 | An alternative definition of "nand" based on imnan 437. See df-nan 1440 for the original definition. This theorem allows various shortenings. (Contributed by Wolf Lammen, 26-Jun-2020.) | ||||||||||||||||||||||
⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓)) | ||||||||||||||||||||||||
Theorem | wl-nancom 32476 | The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Revised by Wolf Lammen, 26-Jun-2020.) | ||||||||||||||||||||||
⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜓 ⊼ 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nannan 32477 | Lemma for handling nested 'nand's. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.) | ||||||||||||||||||||||
⊢ ((𝜑 ⊼ (𝜓 ⊼ 𝜒)) ↔ (𝜑 → (𝜓 ∧ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-nannot 32478 | Show equivalence between negation and the Nicod version. To derive nic-dfneg 1586, apply nanbi 1446. (Contributed by Jeff Hoffman, 19-Nov-2007.) (Revised by Wolf Lammen, 26-Jun-2020.) | ||||||||||||||||||||||
⊢ (¬ 𝜑 ↔ (𝜑 ⊼ 𝜑)) | ||||||||||||||||||||||||
Theorem | wl-nanbi1 32479 | Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.) | ||||||||||||||||||||||
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-nanbi2 32480 | Introduce a left anti-conjunct to both sides of a logical equivalence. (Contributed by SF, 2-Jan-2018.) (Revised by Wolf Lammen, 27-Jun-2020.) | ||||||||||||||||||||||
⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ⊼ 𝜒) ↔ (𝜓 ⊼ 𝜒))) | ||||||||||||||||||||||||
Theorem | wl-naev 32481* | If some set variables can assume different values, then any two distinct set variables cannot always be the same. (Contributed by Wolf Lammen, 10-Aug-2019.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑣) | ||||||||||||||||||||||||
Theorem | wl-hbae1 32482 | This specialization of hbae 2303 does not depend on ax-11 2021. (Contributed by Wolf Lammen, 8-Aug-2021.) | ||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-naevhba1v 32483* | An instance of hbn1w 1960 applied to equality. (Contributed by Wolf Lammen, 7-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-hbnaev 32484* | Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. The latter condition can actually be lifted, but this version is easier to prove. The proof does not use ax-10 2006. (Contributed by Wolf Lammen, 9-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-spae 32485 |
Prove an instance of sp 2041 from ax-13 2234 and Tarski's FOL only, without
distinct variable conditions. The antecedent ∀𝑥𝑥 = 𝑦 holds in a
multi-object universe only if 𝑦 is substituted for 𝑥, or
vice
versa, i.e. both variables are effectively the same. The converse
¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are
distinct, and it so
provides a simple translation of a distinct variable condition to a
logical term. In case studies ∀𝑥𝑥 = 𝑦 and ¬
∀𝑥𝑥 = 𝑦 can
help eliminating distinct variable conditions.
The antecedent ∀𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'. Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2034. Note that this theorem is also provable from ax-12 2034 alone, so you can pick the axiom it is based on. Compare this result to 19.3v 1884 and spaev 1965 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.) | ||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | ||||||||||||||||||||||||
Theorem | wl-cbv3vv 32486* | Avoiding ax-11 2021. (Contributed by Wolf Lammen, 30-Aug-2021.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||||||||||||||||||||||||
Theorem | wl-speqv 32487* | Under the assumption ¬ 𝑥 = 𝑦 a specialized version of sp 2041 is provable from Tarski's FOL and ax13v 2235 only. Note that this reverts the implication in ax13lem1 2236, so in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ 𝑥 = 𝑦 → (∀𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-19.8eqv 32488* | Under the assumption ¬ 𝑥 = 𝑦 a specialized version of 19.8a 2039 is provable from Tarski's FOL and ax13v 2235 only. Note that this reverts the implication in ax13lem2 2284, so in fact (¬ 𝑥 = 𝑦 → (∃𝑥𝑧 = 𝑦 ↔ 𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∃𝑥 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-19.2reqv 32489* | Under the assumption ¬ 𝑥 = 𝑦 the reverse direction of 19.2 1879 is provable from Tarski's FOL and ax13v 2235 only. Note that in conjunction with 19.2 1879 in fact (¬ 𝑥 = 𝑦 → (∀𝑥𝑧 = 𝑦 ↔ ∃𝑥𝑧 = 𝑦)) holds. (Contributed by Wolf Lammen, 17-Apr-2021.) | ||||||||||||||||||||||
⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-dveeq12 32490* | The current form of ax-13 2234 has a particular disadvantage: The condition ¬ 𝑥 = 𝑦 is less versatile than the general form ¬ ∀𝑥𝑥 = 𝑦. You need ax-10 2006 to arrive at the more general form presented here. You need 19.8a 2039 (or ax-12 2034) to restore 𝑦 = 𝑧 from ∃𝑥𝑦 = 𝑧 again. (Contributed by Wolf Lammen, 9-Jun-2021.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||||||||||||||||||||||||
Theorem | wl-nfalv 32491* | If 𝑥 is not present in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Wolf Lammen, 11-Jan-2020.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥∀𝑦𝜑 | ||||||||||||||||||||||||
Theorem | wl-nfimf1 32492 | An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1813 in dvelimdf 2323 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.) | ||||||||||||||||||||||
⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓)) | ||||||||||||||||||||||||
Theorem | wl-nfnbi 32493 | Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 1768 or nfnd 1769. (Contributed by Wolf Lammen, 5-May-2018.) | ||||||||||||||||||||||
⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑) | ||||||||||||||||||||||||
Theorem | wl-nfae1 32494 | Unlike nfae 2304, this specialized theorem avoids ax-11 2021. (Contributed by Wolf Lammen, 26-Jun-2019.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥∀𝑦 𝑦 = 𝑥 | ||||||||||||||||||||||||
Theorem | wl-nfnae1 32495 | Unlike nfnae 2306, this specialized theorem avoids ax-11 2021. (Contributed by Wolf Lammen, 27-Jun-2019.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑥 | ||||||||||||||||||||||||
Theorem | wl-aetr 32496 | A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.) | ||||||||||||||||||||||
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)) | ||||||||||||||||||||||||
Theorem | wl-dral1d 32497 | A version of dral1 2313 with a context. Note: At first glance one might be tempted to generalize this (or a similar) theorem by weakening the first two hypotheses adding a 𝑥 = 𝑦, ∀𝑥𝑥 = 𝑦 or 𝜑 antecedent. wl-equsal1i 32508 and nf5di 2105 show that this is in fact pointless. (Contributed by Wolf Lammen, 28-Jul-2019.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))) | ||||||||||||||||||||||||
Theorem | wl-cbvalnaed 32498 | wl-cbvalnae 32499 with a context. (Contributed by Wolf Lammen, 28-Jul-2019.) | ||||||||||||||||||||||
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜓)) & ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||||||||||||||||||||||||
Theorem | wl-cbvalnae 32499 | A more general version of cbval 2259 when non-free properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf 2322, nfsb2 2348 or dveeq1 2288. (Contributed by Wolf Lammen, 4-Jun-2019.) | ||||||||||||||||||||||
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑) & ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||||||||||||||||||||||||
Theorem | wl-exeq 32500 | The semantics of ∃𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.) | ||||||||||||||||||||||
⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧)) |
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