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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremax13dgen1 2001 Degenerate instance of ax-13 2234 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
𝑥 = 𝑥 → (𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))

Theoremax13dgen2 2002 Degenerate instance of ax-13 2234 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))

Theoremax13dgen3 2003 Degenerate instance of ax-13 2234 where bundled variables 𝑦 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥 𝑦 = 𝑦))

Theoremax13dgen4 2004 Degenerate instance of ax-13 2234 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.)
𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Theoremax13dgen4OLD 2005 Obsolete proof of ax13dgen4 2004 as of 10-Oct-2021. (Contributed by NM, 13-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)

In this section we introduce four additional schemes ax-10 2006, ax-11 2021, ax-12 2034, and ax-13 2234 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "scheme completeness," which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs, and setvar variables ranging over object-language individual variables.

To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax10w 1993, ax11w 1994, ax12w 1997, and ax13w 2000, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all setvar variables to be mutually distinct) can be proved using only the schemes in Tarski's system S2.

An open problem is to show that these four additional schemes are mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 2034 from all others has been shown, and independence of Tarski's ax-6 1875 from all others has been shown; see items 9a and 11 on http://us.metamath.org/award2003.html.

1.5.1  Axiom scheme ax-10 (Quantified Negation)

Axiomax-10 2006 Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 1993) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 21-May-2008.) Use its alias hbn1 2007 instead if you must use it. Any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2006 through ax-13 2234, by invoking ax10w 1993 through ax13w 2000. We encourage proving theorems *without* ax-10 2006 through ax-13 2234 and moving them up to the ax-4 1728 through ax-9 1986 section. (New usage is discouraged.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbn1 2007 Alias for ax-10 2006 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbe1 2008 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by NM, 24-Jan-1993.)
(∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbe1a 2009 Dual statement of hbe1 2008. Modified version of axc7e 2118 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
(∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Theoremnf5-1 2010 One direction of nf5 2102 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)

Theoremnf5i 2011 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑

Theoremnf5dv 2012* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnf5dh 2013 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Oct-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnfe1 2014 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝑥𝜑

Theoremnfa1 2015 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑

Theoremnfna1 2016 A convenience theorem particularly designed to remove dependencies on ax-11 2021 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.)
𝑥 ¬ ∀𝑥𝜑

Theoremnfia1 2017 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥(∀𝑥𝜑 → ∀𝑥𝜓)

Theoremnfnf1 2018 The setvar 𝑥 is not free in 𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 12-Oct-2021.)
𝑥𝑥𝜑

Theoremmodal-5 2019 The analogue in our predicate calculus of axiom (5) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
(¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Theoremnfe1OLD 2020 Obsolete proof of nfe1 2014 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑

1.5.2  Axiom scheme ax-11 (Quantifier Commutation)

Axiomax-11 2021 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 1994) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremalcoms 2022 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
(∀𝑥𝑦𝜑𝜓)       (∀𝑦𝑥𝜑𝜓)

Theoremhbal 2023 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by NM, 12-Mar-1993.)
(𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremalcom 2024 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 30-Jun-1993.)
(∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑)

Theoremalrot3 2025 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦𝑧𝜑 ↔ ∀𝑦𝑧𝑥𝜑)

Theoremalrot4 2026 Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
(∀𝑥𝑦𝑧𝑤𝜑 ↔ ∀𝑧𝑤𝑥𝑦𝜑)

Theoremnfa2 2027 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 18-Oct-2021.)
𝑥𝑦𝑥𝜑

Theoremhbald 2028 Deduction form of bound-variable hypothesis builder hbal 2023. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))

Theoremexcom 2029 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1827, ax-6 1875, ax-7 1922, ax-10 2006, ax-12 2034. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
(∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)

Theoremexcomim 2030 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-5 1827, ax-6 1875, ax-7 1922, ax-10 2006, ax-12 2034. (Revised by Wolf Lammen, 8-Jan-2018.)
(∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)

Theoremexcom13 2031 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)

Theoremexrot3 2032 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)

Theoremexrot4 2033 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)

1.5.3  Axiom scheme ax-12 (Substitution)

Axiomax-12 2034 Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent 𝑥(𝑥 = 𝑦𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2417). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

The original version of this axiom was ax-c15 33192 and was replaced with this shorter ax-12 2034 in Jan. 2007. The old axiom is proved from this one as theorem axc15 2291. Conversely, this axiom is proved from ax-c15 33192 as theorem ax12 2292.

Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 33192) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

See ax12v 2035 and ax12v2 2036 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

This axiom scheme is logically redundant (see ax12w 1997) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.)

(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12v 2035* This is essentially axiom ax-12 2034 weakened by additional restrictions on variables. Besides axc11r 2175, this theorem should be the only one referencing ax-12 2034 directly.

Both restrictions on variables have their own value. If for a moment we assume 𝑦 could be set to 𝑥, then, after elimination of the tautology 𝑥 = 𝑥, immediately we have 𝜑 → ∀𝑥𝜑 for all 𝜑 and 𝑥, that is ax-5 1827, a degenerate result.

The second restriction is not necessary, but a simplification that makes the following interpretation easier to see. Since 𝜑 textually at most depends on 𝑥, we can look at it at some given 'fixed' 𝑦. This theorem now states that the truth value of 𝜑 will stay constant, as long as we 'vary 𝑥 around 𝑦' only such that 𝑥 = 𝑦 still holds. Or in other words, equality is the finest grained logical expression. If you cannot differ two sets by =, you won't find a whatever sophisticated expression that does. One might wonder how the described variation of 𝑥 is possible at all. Note that Metamath is a text processor that easily sees a difference between text chunks {𝑥 ∣ ¬ 𝑥 = 𝑥} and {𝑦 ∣ ¬ 𝑦 = 𝑦}. Our usual interpretation is to abstract from textual variations of the same set, but we are free to interpret Metamath's formalism differently, and in fact let 𝑥 run through all textual representations of sets.

Had we allowed 𝜑 to depend also on 𝑦, this idea is both harder to see, and it is less clear that this extra freedom introduces effects not covered by other axioms. (Contributed by Wolf Lammen, 8-Aug-2020.)

(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12v2 2036* It is possible to remove any restriction on 𝜑 in ax12v 2035. Same as Axiom C8 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2006 and ax-13 2234. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12vOLD 2037* Obsolete proof of ax12v2 2036 as of 24-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2006 and ax-13 2234. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof shortened by Wolf Lammen, 7-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12vOLDOLD 2038* Obsolete proof of ax12v 2035 as of 7-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2006 and ax-13 2234. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorem19.8a 2039 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 1882 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2041. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
(𝜑 → ∃𝑥𝜑)

Theorem19.8aOLD 2040 Obsolete proof of 19.8a 2039. Obsolete as of 21-Dec-2020. Can be deleted as soon as the question of why "MM-PA> min exlimiiv" does not give 19.8a 2039 is answered. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2041. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜑)

Theoremsp 2041 Specialization. A universally quantified wff implies the wff without a quantifier Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). This corresponds to the axiom (T) of modal logic.

For the axiom of specialization presented in many logic textbooks, see theorem stdpc4 2341.

This theorem shows that our obsolete axiom ax-c5 33186 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114.

It appears that this scheme cannot be derived directly from Tarski's axioms without auxiliary axiom scheme ax-12 2034. It is thought the best we can do using only Tarski's axioms is spw 1954. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)

(∀𝑥𝜑𝜑)

Theoremspi 2042 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
𝑥𝜑       𝜑

Theoremsps 2043 Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem2sp 2044 A double specialization (see sp 2041). Another double specialization, closer to PM*11.1, is 2stdpc4 2342. (Contributed by BJ, 15-Sep-2018.)
(∀𝑥𝑦𝜑𝜑)

Theoremspsd 2045 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

Theorem19.2g 2046 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1879 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
(∀𝑥𝜑 → ∃𝑦𝜑)

Theorem19.21bi 2047 Inference form of 19.21 2062 and also deduction form of sp 2041. (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜓)       (𝜑𝜓)

Theorem19.21bbi 2048 Inference removing double quantifier. Version of 19.21bi 2047 with two quanditiers. (Contributed by NM, 20-Apr-1994.)
(𝜑 → ∀𝑥𝑦𝜓)       (𝜑𝜓)

Theorem19.23bi 2049 Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2067. (Contributed by NM, 12-Mar-1993.)
(∃𝑥𝜑𝜓)       (𝜑𝜓)

Theoremnexr 2050 Inference associated with the contrapositive of 19.8a 2039. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑

Theoremqexmid 2051 Quantified excluded middle (see exmid 430). Also known as the drinker paradox (if 𝜑(𝑥) is interpreted as "𝑥 drinks", then this theorem tells that there exists a person such that, if this person drinks, then everyone drinks). Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
𝑥(𝜑 → ∀𝑥𝜑)

Theoremnf5r 2052 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))

Theoremnf5ri 2053 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       (𝜑 → ∀𝑥𝜑)

Theoremnf5rd 2054 Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))

Theoremnfim1 2055 A closed form of nfim 1813. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theoremnfan1 2056 A closed form of nfan 1816. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theorem19.3 2057 A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1884 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥𝜑𝜑)

Theorem19.9d 2058 A deduction version of one direction of 19.9 2060. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))

Theorem19.9t 2059 A closed version of 19.9 2060. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
(Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))

Theorem19.9 2060 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1883 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
𝑥𝜑       (∃𝑥𝜑𝜑)

Theorem19.21t 2061 Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2062. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorem19.21 2062 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 1855 for a version requiring fewer axioms. See also 19.21h 2107. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theoremstdpc5 2063 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis 𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 1927. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1854 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theoremstdpc5OLD 2064 Obsolete proof of stdpc5 2063 as of 11-Oct-2021. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2006. (Revised by Wolf Lammen, 4-Jul-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem19.21-2 2065 Version of 19.21 2062 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))

Theorem19.23t 2066 Closed form of Theorem 1977.23 of [Margaris] p. 90. See 19.23 2067. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorem19.23 2067 Theorem 19.23 of [Margaris] p. 90. See 19.23v 1889 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremalimd 2068 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1729. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalrimi 2069 Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2062. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalrimdd 2070 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2062. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremalrimd 2071 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2062. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremeximd 2072 Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1751. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremexlimi 2073 Inference associated with 19.23 2067. See exlimiv 1845 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜓    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremexlimd 2074 Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremexlimdd 2075 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

Theoremnexd 2076 Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremalbid 2077 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbid 2078 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremnfbidf 2079 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Theorem19.16 2080 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Theorem19.17 2081 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorem19.27 2082 Theorem 19.27 of [Margaris] p. 90. See 19.27v 1895 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28 2083 Theorem 19.28 of [Margaris] p. 90. See 19.28v 1896 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.19 2084 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Theorem19.36 2085 Theorem 19.36 of [Margaris] p. 90. See 19.36v 1891 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.36i 2086 Inference associated with 19.36 2085. See 19.36iv 1892 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
𝑥𝜓    &   𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem19.37 2087 Theorem 19.37 of [Margaris] p. 90. See 19.37v 1897 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Theorem19.32 2088 Theorem 19.32 of [Margaris] p. 90. See 19.32v 1856 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Theorem19.31 2089 Theorem 19.31 of [Margaris] p. 90. See 19.31v 1857 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.41 2090 Theorem 19.41 of [Margaris] p. 90. See 19.41v 1901 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.42-1 2091 One direction of 19.42 2092. (Contributed by Wolf Lammen, 10-Jul-2021.)
𝑥𝜑       ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.42 2092 Theorem 19.42 of [Margaris] p. 90. See 19.42v 1905 for a version requiring fewer axioms. See exan 1775 for an immediate version. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theorem19.44 2093 Theorem 19.44 of [Margaris] p. 90. See 19.44v 1899 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45 2094 Theorem 19.45 of [Margaris] p. 90. See 19.45v 1900 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theoremequsexv 2095* Version of equsex 2281 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremsbequ1 2096 An equality theorem for substitution. (Contributed by NM, 16-May-1993.)
(𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))

Theoremsbequ12 2097 An equality theorem for substitution. (Contributed by NM, 14-May-1993.)
(𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))

Theoremsbequ12r 2098 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Theoremsbequ12a 2099 An equality theorem for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Jun-2019.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

Theoremsbid 2100 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)
([𝑥 / 𝑥]𝜑𝜑)

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