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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.27v 2001* Version of 19.27 2229 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
Theorem19.28v 2002* Version of 19.28 2230 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
 
Theorem19.37v 2003* Version of 19.37 2234 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
 
Theorem19.44v 2004* Version of 19.44 2239 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.45v 2005* Version of 19.45 2240 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
 
Theoremspimevw 2006* Existential introduction, using implicit substitution. This is to spimew 1979 what spimvw 2007 is to spimw 1978. Version of spimev 2393 and spimefv 2200 with an additional disjoint variable condition, using only Tarski's FOL axiom schemes. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 17-Mar-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
Theoremspimvw 2007* A weak form of specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. For stronger forms using more axioms, see spimv 2391 and spimfv 2241. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspvv 2008* Specialization, using implicit substitution. Version of spv 2394 with a disjoint variable condition, which does not require ax-7 2020, ax-12 2179, ax-13 2373. (Contributed by NM, 30-Aug-1993.) (Revised by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremspfalw 2009 Version of sp 2184 when 𝜑 is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
¬ 𝜑       (∀𝑥𝜑𝜑)
 
Theoremchvarvv 2010* Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvarv 2397 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 20-Apr-1994.) (Revised by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremequs4v 2011* Version of equs4 2417 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-May-1993.) (Revised by BJ, 31-May-2019.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremalequexv 2012* Version of equs4v 2011 with its consequence simplified by exsimpr 1876. (Contributed by BJ, 9-Nov-2021.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
 
Theoremexsbim 2013* One direction of the equivalence in exsb 2361 is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
(∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
 
Theoremequsv 2014* If a formula does not contain a variable 𝑥, then it is equivalent to the corresponding prototype of substitution with a fresh variable (see sb6 2095). (Contributed by BJ, 23-Jul-2023.)
(∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
 
Theoremequsalvw 2015* Version of equsalv 2268 with a disjoint variable condition, and of equsal 2418 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2016. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremequsexvw 2016* Version of equsexv 2269 with a disjoint variable condition, and of equsex 2419 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2015. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
TheoremequsexvwOLD 2017* Obsolete version of equsexvw 2016 as of 23-Oct-2023. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremcbvaliw 2018* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)
 
Theoremcbvalivw 2019* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)
 
1.4.7  Axiom scheme ax-7 (Equality)
 
Axiomax-7 2020 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is a right-Euclidean binary relation (this is similar, but not identical, to being transitive, which is proved as equtr 2033). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle".

We prove in ax7 2028 that this axiom can be recovered from its weakened version ax7v 2021 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 2020 should be ax7v 2021. See the comment of ax7v 2021 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 2028 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 
Theoremax7v 2021* Weakened version of ax-7 2020, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 2020, and it should be referenced only by its two weakened versions ax7v1 2022 and ax7v2 2023, from which ax-7 2020 is then rederived as ax7 2028, which shows that either ax7v 2021 or the conjunction of ax7v1 2022 and ax7v2 2023 is sufficient.

In ax7v 2021, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2021 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 2026 and equid 2024 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2028 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 
Theoremax7v1 2022* First of two weakened versions of ax7v 2021, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 
Theoremax7v2 2023* Second of two weakened versions of ax7v 2021, with an extra disjoint variable condition on 𝑦, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 
Theoremequid 2024 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
𝑥 = 𝑥
 
Theoremnfequid 2025 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
𝑦 𝑥 = 𝑥
 
Theoremequcomiv 2026* Weaker form of equcomi 2029 with a disjoint variable condition on 𝑥, 𝑦. This is an intermediate step and equcomi 2029 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦𝑦 = 𝑥)
 
Theoremax6evr 2027* A commuted form of ax6ev 1977. (Contributed by BJ, 7-Dec-2020.)
𝑥 𝑦 = 𝑥
 
Theoremax7 2028 Proof of ax-7 2020 from ax7v1 2022 and ax7v2 2023 (and earlier axioms), proving sufficiency of the conjunction of the latter two weakened versions of ax7v 2021, which is itself a weakened version of ax-7 2020.

Note that the weakened version of ax-7 2020 obtained by adding a disjoint variable condition on 𝑥, 𝑧 (resp. on 𝑦, 𝑧) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 
Theoremequcomi 2029 Commutative law for equality. Equality is a symmetric relation. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)
(𝑥 = 𝑦𝑦 = 𝑥)
 
Theoremequcom 2030 Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)
 
Theoremequcomd 2031 Deduction form of equcom 2030, symmetry of equality. For the versions for classes, see eqcom 2746 and eqcomd 2745. (Contributed by BJ, 6-Oct-2019.)
(𝜑𝑥 = 𝑦)       (𝜑𝑦 = 𝑥)
 
Theoremequcoms 2032 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦𝜑)       (𝑦 = 𝑥𝜑)
 
Theoremequtr 2033 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))
 
Theoremequtrr 2034 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
 
Theoremequeuclr 2035 Commuted version of equeucl 2036 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))
 
Theoremequeucl 2036 Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2020.) Curried (exported) form of equtr2 2039. (Contributed by BJ, 11-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))
 
Theoremequequ1 2037 An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
 
Theoremequequ2 2038 An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) (Proof shortened by BJ, 12-Apr-2021.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
 
Theoremequtr2 2039 Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2036. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.)
((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
 
Theoremstdpc6 2040 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 2252.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
𝑥 𝑥 = 𝑥
 
Theoremequvinv 2041* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2145, ax-13 2373. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
(𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
 
Theoremequvinva 2042* A modified version of the forward implication of equvinv 2041 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))
 
Theoremequvelv 2043* A biconditional form of equvel 2457 with disjoint variable conditions and proved from Tarski's FOL axiom schemes. (Contributed by Andrew Salmon, 2-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
(∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ 𝑥 = 𝑦)
 
Theoremax13b 2044 An equivalence between two ways of expressing ax-13 2373. See the comment for ax-13 2373. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)
((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑))))
 
Theoremspfw 2045* Weak version of sp 2184. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)
 
Theoremspw 2046* Weak version of the specialization scheme sp 2184. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2184 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2184 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2139 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2184 are spfw 2045 (minimal distinct variable requirements), spnfw 1989 (when 𝑥 is not free in ¬ 𝜑), spvw 1990 (when 𝑥 does not appear in 𝜑), sptruw 1813 (when 𝜑 is true), spfalw 2009 (when 𝜑 is false), and spvv 2008 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)
 
Theoremcbvalw 2047* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
Theoremcbvalvw 2048* Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvalv 2401 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
Theoremcbvexvw 2049* Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvexv 2402 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
Theoremcbvaldvaw 2050* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 2410 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
Theoremcbvexdvaw 2051* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 2411 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 
Theoremcbval2vw 2052* Rule used to change bound variables, using implicit substitution. Version of cbval2vv 2414 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
 
Theoremcbvex2vw 2053* Rule used to change bound variables, using implicit substitution. Version of cbvex2vv 2415 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
 
Theoremcbvex4vw 2054* Rule used to change bound variables, using implicit substitution. Version of cbvex4v 2416 with more disjoint variable conditions, which requires fewer axioms. (Contributed by NM, 26-Jul-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
 
Theoremalcomiw 2055* Weak version of alcom 2164. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Dec-2023.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
TheoremalcomiwOLD 2056* Obsolete version of alcomiw 2055 as of 28-Dec-2023. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 12-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremhbn1fw 2057* Weak version of ax-10 2145 from which we can prove any ax-10 2145 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhbn1w 2058* Weak version of hbn1 2146. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhba1w 2059* Weak version of hba1 2298. See comments for ax10w 2133. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremhbe1w 2060* Weak version of hbe1 2147. See comments for ax10w 2133. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
 
Theoremhbalw 2061* Weak version of hbal 2175. Uses only Tarski's FOL axiom schemes. Unlike hbal 2175, this theorem requires that 𝑥 and 𝑦 be distinct, i.e., not be bundled. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)
 
Theoremspaev 2062* A special instance of sp 2184 applied to an equality with a disjoint variable condition. Unlike the more general sp 2184, we can prove this without ax-12 2179. Instance of aeveq 2066.

The antecedent 𝑥𝑥 = 𝑦 with distinct 𝑥 and 𝑦 is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition 𝑥𝑥 = 𝑦 is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
 
Theoremcbvaev 2063* Change bound variable in an equality with a disjoint variable condition. Instance of aev 2067. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)
 
Theoremaevlem0 2064* Lemma for aevlem 2065. Instance of aev 2067. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2179. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 30-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
 
Theoremaevlem 2065* Lemma for aev 2067 and axc16g 2261. Change free and bound variables. Instance of aev 2067. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2373, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)
 
Theoremaeveq 2066* The antecedent 𝑥𝑥 = 𝑦 with a disjoint variable condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦𝑧 = 𝑡)
 
Theoremaev 2067* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2162. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2373, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2179. (Revised by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢)
 
Theoremaev2 2068* A version of aev 2067 with two universal quantifiers in the consequent. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 2066, aev 2067, aev2 2068).

Using aev 2067 and alrimiv 1934, one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors , , , , , =, , , ∃*, ∃!, . An example is given by aevdemo 28410. This list cannot be extended to ¬ or since the scheme 𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1802, ax-1 6-- ax-13 2373 (as the one-element universe shows).

(Contributed by BJ, 23-Mar-2021.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)
 
Theoremhbaev 2069* All variables are effectively bound in an identical variable specifier. Version of hbae 2432 with a disjoint variable condition, requiring fewer axioms. Instance of aev2 2068. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 22-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
 
Theoremnaev 2070* 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)
 
Theoremnaev2 2071* Generalization of hbnaev 2072. (Contributed by Wolf Lammen, 9-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢)
 
Theoremhbnaev 2072* Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. This condition is dropped in hbnae 2433, at the expense of more axiom dependencies. Instance of naev2 2071. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 9-Apr-2021.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
 
1.4.8  Define proper substitution
 
Theoremsbjust 2073* Justification theorem for df-sb 2075 proved from Tarski's FOL axiom schemes. (Contributed by BJ, 22-Jan-2023.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))
 
Syntaxwsb 2074 Extend wff definition to include proper substitution (read "the wff that results when 𝑦 is properly substituted for 𝑥 in wff 𝜑"). (Contributed by NM, 24-Jan-2006.)
wff [𝑦 / 𝑥]𝜑
 
Definitiondf-sb 2075* Define proper substitution. For our notation, we use [𝑡 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑡 for 𝑥 in the wff 𝜑". That is, 𝑡 properly replaces 𝑥. For example, [𝑡 / 𝑥]𝑧𝑥 is the same as 𝑧𝑡 (when 𝑥 and 𝑧 are distinct), as shown in elsb4 2130.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑡) is the wff that results when 𝑡 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑡, then 𝜑(𝑡) is 𝑡 = 𝑡, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem.

A very similar notation, namely (𝑦𝑥)𝜑, was introduced in Bourbaki's Set Theory (Chapter 1, Description of Formal Mathematic, 1953).

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see Theorems sbequ 2093, sbcom2 2169 and sbid2v 2514).

Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2257 shows. We achieve this by applying twice Tarski's definition sb6 2095 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2284 with respect to sb5 2276. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2486 shows. Another version that mixes free and bound variables is dfsb3 2499. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2276 and sb6 2095.

Note that the occurrences of a given variable in the definiens are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a disjoint variable condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. (Contributed by NM, 10-May-1993.) Revised from the original definition dfsb1 2486. (Revised by BJ, 22-Dec-2020.)

([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremsbt 2076 A substitution into a theorem yields a theorem. See sbtALT 2079 for a shorter proof requiring more axioms. See chvar 2396 and chvarv 2397 for versions using implicit substitution. (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.) Revise df-sb 2075. (Revised by Steven Nguyen, 6-Jul-2023.)
𝜑       [𝑡 / 𝑥]𝜑
 
Theoremsbtru 2077 The result of substituting in the truth constant "true" is true. (Contributed by BJ, 2-Sep-2023.)
[𝑦 / 𝑥]⊤
 
Theoremstdpc4 2078 The specialization axiom of standard predicate calculus. It states that if a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑡 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "𝑥𝜑(𝑥) → 𝜑(𝑡), provided that 𝑡 is free for 𝑥 in 𝜑(𝑥)". Axiom 4 of [Mendelson] p. 69. See also spsbc 3698 and rspsbc 3780. (Contributed by NM, 14-May-1993.) Revise df-sb 2075. (Revised by BJ, 22-Dec-2020.)
(∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
 
TheoremsbtALT 2079 Alternate proof of sbt 2076, shorter but using ax-4 1816 and ax-5 1917. (Contributed by NM, 21-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑       [𝑦 / 𝑥]𝜑
 
Theorem2stdpc4 2080 A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2078 for the analogous single specialization. See 2sp 2187 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)
(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
 
Theoremsbi1 2081 Distribute substitution over implication. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 
Theoremspsbim 2082 Distribute substitution over implication. Closed form of sbimi 2084. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2075. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)
(∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
 
Theoremspsbbi 2083 Biconditional property for substitution. Closed form of sbbii 2086. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2075. (Revised by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))
 
Theoremsbimi 2084 Distribute substitution over implication. (Contributed by NM, 25-Jun-1998.) Revise df-sb 2075. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)
(𝜑𝜓)       ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)
 
Theoremsb2imi 2085 Distribute substitution over implication. Compare al2imi 1822. (Contributed by Steven Nguyen, 13-Aug-2023.)
(𝜑 → (𝜓𝜒))       ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
 
Theoremsbbii 2086 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)
(𝜑𝜓)       ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)
 
Theorem2sbbii 2087 Infer double substitution into both sides of a logical equivalence. (Contributed by AV, 30-Jul-2023.)
(𝜑𝜓)       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
 
Theoremsbimdv 2088* Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2246. (Contributed by Wolf Lammen, 6-May-2023.) Revise df-sb 2075. (Revised by Steven Nguyen, 6-Jul-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
 
Theoremsbbidv 2089* Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 2247. (Contributed by Wolf Lammen, 6-May-2023.) (Proof shortened by Steven Nguyen, 6-Jul-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑥]𝜒))
 
Theoremsban 2090 Conjunction inside and outside of a substitution are equivalent. Compare 19.26 1877. (Contributed by NM, 14-May-1993.) (Proof shortened by Steven Nguyen, 13-Aug-2023.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
 
Theoremsb3an 2091 Threefold conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
([𝑦 / 𝑥](𝜑𝜓𝜒) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓 ∧ [𝑦 / 𝑥]𝜒))
 
Theoremspsbe 2092 Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2075. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.)
([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
 
Theoremsbequ 2093 Equality property for substitution, from Tarski's system. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.) Revise df-sb 2075. (Revised by BJ, 30-Dec-2020.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
 
Theoremsbequi 2094 An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof shortened by Steven Nguyen, 7-Jul-2023.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
 
Theoremsb6 2095* Alternate definition of substitution when variables are disjoint. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" also holds without a disjoint variable condition (sb2 2481). Theorem sb6f 2502 replaces the disjoint variable condition with a nonfreeness hypothesis. Theorem sb4b 2476 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2075. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2162. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.)
([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑))
 
Theorem2sb6 2096* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
 
Theoremsb1v 2097* One direction of sb5 2276, provable from fewer axioms. Version of sb1 2480 with a disjoint variable condition using fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 20-Jan-2024.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremsbv 2098* Substitution for a variable not occurring in a proposition. See sbf 2271 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2098 can be proved directly by chaining equsv 2014 with sb6 2095. (Contributed by BJ, 22-Dec-2020.)
([𝑡 / 𝑥]𝜑𝜑)
 
Theoremsbcom4 2099* Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2100 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)
([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 
Theorempm11.07 2100 Axiom *11.07 in [WhiteheadRussell] p. 159. The original reads: *11.07 "Whatever possible argument 𝑥 may be, 𝜑(𝑥, 𝑦) is true whatever possible argument 𝑦 may be" implies the corresponding statement with 𝑥 and 𝑦 interchanged except in "𝜑(𝑥, 𝑦)". Under our formalism this appears to correspond to idi 1 and not to sbcom4 2099 as earlier thought. See https://groups.google.com/g/metamath/c/iS0fOvSemC8/m/M1zTH8wxCAAJ 2099. (Contributed by BJ, 16-Sep-2018.) (New usage is discouraged.)
𝜑       𝜑
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46010
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