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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hbab 2601* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | ||
Theorem | nfsab 2602* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | ||
Definition | df-cleq 2603* |
Define the equality connective between classes. Definition 2.7 of
[Quine] p. 18. Also Definition 4.5 of
[TakeutiZaring] p. 13; Chapter 4
provides its justification and methods for eliminating it. Note that
its elimination will not necessarily result in a single wff in the
original language but possibly a "scheme" of wffs.
This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce 𝑦 = 𝑧 ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2594). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated. We could avoid this complication by introducing a new symbol, say =2, in place of =. This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition. One of our theorems would then be 𝑥 =2 𝑦 ↔ 𝑥 = 𝑦 by invoking Extensionality. However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality. See also comments under df-clab 2597, df-clel 2606, and abeq2 2719. In the form of dfcleq 2604, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. While the three class definitions df-clab 2597, df-cleq 2603, and df-clel 2606 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 15-Sep-1993.) |
⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | dfcleq 2604* | The same as df-cleq 2603 with the hypothesis removed using the Axiom of Extensionality ax-ext 2590. (Contributed by NM, 15-Sep-1993.) |
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | cvjust 2605* | Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1474, which allows us to substitute a setvar variable for a class variable. See also cab 2596 and df-clab 2597. Note that this is not a rigorous justification, because cv 1474 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.) |
⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} | ||
Definition | df-clel 2606* |
Define the membership connective between classes. Theorem 6.3 of
[Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we
adopt as a definition. See these references for its metalogical
justification. Note that like df-cleq 2603 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2603 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 1985), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2597. Alternate definitions of 𝐴 ∈ 𝐵 (but that
require either 𝐴 or 𝐵 to be a set) are shown
by clel2 3309,
clel3 3311, and clel4 3312.
This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. While the three class definitions df-clab 2597, df-cleq 2603, and df-clel 2606 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker. For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.) |
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | ||
Theorem | eqriv 2607* | Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | eqrdv 2608* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | eqrdav 2609* | Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | eqid 2610 |
Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
p. 41.
This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also biid 250. An early mention of this law can be found in Aristotle, Metaphysics, Z.17, 1041a10-20. (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Oct-2017.) |
⊢ 𝐴 = 𝐴 | ||
Theorem | eqidd 2611 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
⊢ (𝜑 → 𝐴 = 𝐴) | ||
Theorem | eqeq1d 2612 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq1dALT 2613 | Shorter proof of eqeq1d 2612 based on more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq1 2614 | Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | eqeq1i 2615 | Inference from equality to equivalence of equalities. (Contributed by NM, 15-Jul-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐶) | ||
Theorem | eqcomd 2616 | Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) Allow shortening of eqcom 2617. (Revised by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 = 𝐴) | ||
Theorem | eqcom 2617 | Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | ||
Theorem | eqcoms 2618 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 24-Jun-1993.) |
⊢ (𝐴 = 𝐵 → 𝜑) ⇒ ⊢ (𝐵 = 𝐴 → 𝜑) | ||
Theorem | eqcomi 2619 | Inference from commutative law for class equality. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 𝐵 = 𝐴 | ||
Theorem | eqeq2d 2620 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Allow shortening of eqeq2 2621. (Revised by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) | ||
Theorem | eqeq2 2621 | Equality implies equivalence of equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) | ||
Theorem | eqeq2i 2622 | Inference from equality to equivalence of equalities. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵) | ||
Theorem | eqeq12 2623 | Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeq12i 2624 | A useful inference for substituting definitions into an equality. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷) | ||
Theorem | eqeq12d 2625 | A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeqan12d 2626 | A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2627. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeqan12dALT 2627 | Alternate proof of eqeqan12d 2626. This proof has one more step but one fewer essential step. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqeqan12rd 2628 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | ||
Theorem | eqtr 2629 | Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶) | ||
Theorem | eqtr2 2630 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶) | ||
Theorem | eqtr3 2631 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵) | ||
Theorem | eqtri 2632 | An equality transitivity inference. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 = 𝐶 | ||
Theorem | eqtr2i 2633 | An equality transitivity inference. (Contributed by NM, 21-Feb-1995.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐶 = 𝐴 | ||
Theorem | eqtr3i 2634 | An equality transitivity inference. (Contributed by NM, 6-May-1994.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 = 𝐶 ⇒ ⊢ 𝐵 = 𝐶 | ||
Theorem | eqtr4i 2635 | An equality transitivity inference. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 = 𝐶 | ||
Theorem | 3eqtri 2636 | An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐴 = 𝐷 | ||
Theorem | 3eqtrri 2637 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 = 𝐶 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐷 = 𝐴 | ||
Theorem | 3eqtr2i 2638 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐴 = 𝐷 | ||
Theorem | 3eqtr2ri 2639 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 𝐷 = 𝐴 | ||
Theorem | 3eqtr3i 2640 | An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶 = 𝐷 | ||
Theorem | 3eqtr3ri 2641 | An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐷 = 𝐶 | ||
Theorem | 3eqtr4i 2642 | An inference from three chained equalities. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐶 = 𝐷 | ||
Theorem | 3eqtr4ri 2643 | An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐷 = 𝐶 | ||
Theorem | eqtrd 2644 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | eqtr2d 2645 | An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | eqtr3d 2646 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | eqtr4d 2647 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | 3eqtrd 2648 | A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐴 = 𝐷) | ||
Theorem | 3eqtrrd 2649 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 = 𝐴) | ||
Theorem | 3eqtr2d 2650 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐴 = 𝐷) | ||
Theorem | 3eqtr2rd 2651 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 = 𝐴) | ||
Theorem | 3eqtr3d 2652 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr3rd 2653 | A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 = 𝐶) | ||
Theorem | 3eqtr4d 2654 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4rd 2655 | A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐷 = 𝐶) | ||
Theorem | syl5eq 2656 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl5req 2657 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl5eqr 2658 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl5reqr 2659 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl6eq 2660 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl6req 2661 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | syl6eqr 2662 | An equality transitivity deduction. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | syl6reqr 2663 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐶 = 𝐴) | ||
Theorem | sylan9eq 2664 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | ||
Theorem | sylan9req 2665 | An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | ||
Theorem | sylan9eqr 2666 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐵 = 𝐶) ⇒ ⊢ ((𝜓 ∧ 𝜑) → 𝐴 = 𝐶) | ||
Theorem | 3eqtr3g 2667 | A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr3a 2668 | A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4g 2669 | A chained equality inference, useful for converting to definitions. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | 3eqtr4a 2670 | A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐶 = 𝐴) & ⊢ (𝜑 → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐶 = 𝐷) | ||
Theorem | eq2tri 2671 | A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.) |
⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹) & ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺) ⇒ ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺)) | ||
Theorem | eleq1d 2672 | Deduction from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.) Allow shortening of eleq1 2676. (Revised by Wolf Lammen, 20-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | ||
Theorem | eleq2d 2673 | Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq2dOLD 2674 | Obsolete proof of eleq2d 2673 as of 16-Nov-2020. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq2dALT 2675 | Alternate proof of eleq2d 2673, shorter at the expense of using more axioms. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq1 2676 | Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | ||
Theorem | eleq2 2677 | Equality implies equivalence of membership. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 20-Nov-2019.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵)) | ||
Theorem | eleq12 2678 | Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | eleq1i 2679 | Inference from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) | ||
Theorem | eleq2i 2680 | Inference from equality to equivalence of membership. (Contributed by NM, 26-May-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵) | ||
Theorem | eleq12i 2681 | Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷) | ||
Theorem | eleq12d 2682 | Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) | ||
Theorem | eleq1a 2683 | A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.) |
⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵)) | ||
Theorem | eqeltri 2684 | Substitution of equal classes into membership relation. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eqeltrri 2685 | Substitution of equal classes into membership relation. (Contributed by NM, 21-Jun-1993.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐵 ∈ 𝐶 | ||
Theorem | eleqtri 2686 | Substitution of equal classes into membership relation. (Contributed by NM, 15-Jul-1993.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eleqtrri 2687 | Substitution of equal classes into membership relation. (Contributed by NM, 15-Jul-1993.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴 ∈ 𝐶 | ||
Theorem | eqeltrd 2688 | Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eqeltrrd 2689 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝐶) | ||
Theorem | eleqtrd 2690 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | eleqtrrd 2691 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eqel 2692 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eqelr 2693 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eleq 2694 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl5eleqr 2695 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eqel 2696 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eqelr 2697 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eleq 2698 | A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | syl6eleqr 2699 | A membership and equality inference. (Contributed by NM, 24-Apr-2005.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐶) | ||
Theorem | 3eltr3i 2700 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
⊢ 𝐴 ∈ 𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶 ∈ 𝐷 |
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