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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | equsalvw 2001* | Version of equsalv 2259 with a disjoint variable condition, and of equsal 2433 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2002. (Contributed by BJ, 31-May-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equsexvw 2002* | Version of equsexv 2260 with a disjoint variable condition, and of equsex 2434 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2001. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | equsexvwOLD 2003* | Obsolete version of equsexvw 2002 as of 23-Oct-2023. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | cbvaliw 2004* | 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 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | cbvalivw 2005* | 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 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Axiom | ax-7 2006 |
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 2019). 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 2014 that this axiom can be recovered from its weakened version ax7v 2007 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 2006 should be ax7v 2007. See the comment of ax7v 2007 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 2014 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Theorem | ax7v 2007* |
Weakened version of ax-7 2006, with a disjoint variable condition on
𝑥,
𝑦. This should be
the only proof referencing ax-7 2006, and it
should be referenced only by its two weakened versions ax7v1 2008 and
ax7v2 2009, from which ax-7 2006
is then rederived as ax7 2014, which shows
that either ax7v 2007 or the conjunction of ax7v1 2008 and ax7v2 2009 is
sufficient.
In ax7v 2007, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2007 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥 → 𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 2012 and equid 2010 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2014 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Theorem | ax7v1 2008* | First of two weakened versions of ax7v 2007, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Theorem | ax7v2 2009* | Second of two weakened versions of ax7v 2007, with an extra disjoint variable condition on 𝑦, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Theorem | equid 2010 | 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 | ||
Theorem | nfequid 2011 | 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.) |
⊢ Ⅎ𝑦 𝑥 = 𝑥 | ||
Theorem | equcomiv 2012* | Weaker form of equcomi 2015 with a disjoint variable condition on 𝑥, 𝑦. This is an intermediate step and equcomi 2015 is fully recovered later. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | ||
Theorem | ax6evr 2013* | A commuted form of ax6ev 1963. (Contributed by BJ, 7-Dec-2020.) |
⊢ ∃𝑥 𝑦 = 𝑥 | ||
Theorem | ax7 2014 |
Proof of ax-7 2006 from ax7v1 2008 and ax7v2 2009 (and earlier axioms), proving
sufficiency of the conjunction of the latter two weakened versions of
ax7v 2007, which is itself a weakened version of ax-7 2006.
Note that the weakened version of ax-7 2006 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 = 𝑧 → 𝑦 = 𝑧)) | ||
Theorem | equcomi 2015 | 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 = 𝑦 → 𝑦 = 𝑥) | ||
Theorem | equcom 2016 | Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.) |
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | ||
Theorem | equcomd 2017 | Deduction form of equcom 2016, symmetry of equality. For the versions for classes, see eqcom 2828 and eqcomd 2827. (Contributed by BJ, 6-Oct-2019.) |
⊢ (𝜑 → 𝑥 = 𝑦) ⇒ ⊢ (𝜑 → 𝑦 = 𝑥) | ||
Theorem | equcoms 2018 | An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.) |
⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (𝑦 = 𝑥 → 𝜑) | ||
Theorem | equtr 2019 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | ||
Theorem | equtrr 2020 | 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 = 𝑥 → 𝑧 = 𝑦)) | ||
Theorem | equeuclr 2021 | Commuted version of equeucl 2022 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.) |
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥)) | ||
Theorem | equeucl 2022 | Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2006.) Curried (exported) form of equtr2 2025. (Contributed by BJ, 11-Apr-2021.) |
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | ||
Theorem | equequ1 2023 | An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | ||
Theorem | equequ2 2024 | 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 = 𝑥 ↔ 𝑧 = 𝑦)) | ||
Theorem | equtr2 2025 | Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2022. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) | ||
Theorem | stdpc6 2026 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 2243.) 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 | ||
Theorem | equvinv 2027* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2136, ax-13 2383. (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 = 𝑥 ∧ 𝑧 = 𝑦)) | ||
Theorem | equvinva 2028* | A modified version of the forward implication of equvinv 2027 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.) |
⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) | ||
Theorem | equvelv 2029* | A biconditional form of equvel 2474 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 = 𝑥 → 𝑧 = 𝑦) ↔ 𝑥 = 𝑦) | ||
Theorem | ax13b 2030 | An equivalence between two ways of expressing ax-13 2383. See the comment for ax-13 2383. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.) |
⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑)))) | ||
Theorem | spfw 2031* | Weak version of sp 2172. 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 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | spw 2032* | Weak version of the specialization scheme sp 2172. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2172 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2172 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2130 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2172 are spfw 2031 (minimal distinct variable requirements), spnfw 1975 (when 𝑥 is not free in ¬ 𝜑), spvw 1976 (when 𝑥 does not appear in 𝜑), sptruw 1798 (when 𝜑 is true), and spfalw 1995 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | cbvalw 2033* | Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) & ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) & ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvalvw 2034* | Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvalv 2412 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvexvw 2035* | Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvexv 2413 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 19-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | cbvaldvaw 2036* | Version of cbvaldva 2424 with a disjoint variable condition, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | cbvexdvaw 2037* | Version of cbvexdva 2425 with a disjoint variable condition, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | cbval2vw 2038* | Version of cbval2vv 2429 with more disjoint variable conditions, which requires fewer axioms . (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
Theorem | cbvex2vw 2039* | Version of cbvex2vv 2430 with more disjoint variable conditions, which requires fewer axioms . (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | ||
Theorem | cbvex4vw 2040* | Version of cbvex4v 2431 with more disjoint variable conditions, which requires fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) & ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) | ||
Theorem | alcomiw 2041* | Weak version of alcom 2153. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Dec-2023.) |
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | alcomiwOLD 2042* | Obsolete version of alcomiw 2041 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 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | ||
Theorem | hbn1fw 2043* | Weak version of ax-10 2136 from which we can prove any ax-10 2136 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 → ∀𝑦∀𝑥𝜑) & ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) & ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) & ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) & ⊢ (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hbn1w 2044* | Weak version of hbn1 2137. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hba1w 2045* | Weak version of hba1 2293. See comments for ax10w 2124. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | hbe1w 2046* | Weak version of hbe1 2138. See comments for ax10w 2124. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | ||
Theorem | hbalw 2047* | Weak version of hbal 2164. Uses only Tarski's FOL axiom schemes. Unlike hbal 2164, this theorem requires that 𝑥 and 𝑦 be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) | ||
Theorem | spaev 2048* |
A special instance of sp 2172 applied to an equality with a disjoint
variable condition. Unlike the more general sp 2172, we
can prove this
without ax-12 2167. Instance of aeveq 2052.
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 = 𝑦 → 𝑥 = 𝑦) | ||
Theorem | cbvaev 2049* | Change bound variable in an equality with a disjoint variable condition. Instance of aev 2053. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦) | ||
Theorem | aevlem0 2050* | Lemma for aevlem 2051. Instance of aev 2053. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2167. (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) | ||
Theorem | aevlem 2051* | Lemma for aev 2053 and axc16g 2252. Change free and bound variables. Instance of aev 2053. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2383, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡) | ||
Theorem | aeveq 2052* | 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 = 𝑦 → 𝑧 = 𝑡) | ||
Theorem | aev 2053* | A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2151. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2383, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2167. (Revised by Wolf Lammen, 19-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢) | ||
Theorem | aev2 2054* |
A version of aev 2053 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 2052, aev 2053,
aev2 2054).
Using aev 2053 and alrimiv 1919, 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 →, ↔, ∧, ∨, ⊤, =, ∀, ∃, ∃*, ∃!, Ⅎ. An example is given by aevdemo 28167. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1787, ax-1 6-- ax-13 2383 (as the one-element universe shows). (Contributed by BJ, 23-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣) | ||
Theorem | hbaev 2055* | Version of hbae 2448 with a disjoint variable condition, requiring fewer axioms. Instance of aev2 2054. (Contributed by Wolf Lammen, 22-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) | ||
Theorem | naev 2056* | 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 | naev2 2057* | Generalization of hbnaev 2058. (Contributed by Wolf Lammen, 9-Apr-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢) | ||
Theorem | hbnaev 2058* | Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. This condition is dropped in hbnae 2449, at the expense of more axiom dependencies. Instance of naev2 2057. (Contributed by Wolf Lammen, 9-Apr-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | sbjust 2059* | Justification theorem for df-sb 2061 proved from Tarski's FOL. (Contributed by BJ, 22-Jan-2023.) |
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | ||
Syntax | wsb 2060 | 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 | ||
Definition | df-sb 2061* |
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]𝑧 ∈ 𝑥 is the same as 𝑧 ∈ 𝑡 (when 𝑥
and 𝑧 are distinct), as shown in elsb4 2121.
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 (𝑦 ∣ 𝑥)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 2081, sbcom2 2158 and sbid2v 2547). Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2248 shows. We achieve this by applying twice Tarski's definition sb6 2084 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2277 with respect to sb5 2268. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2506 shows. Another version that mixes free and bound variables is dfsb3 2529. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2268 and sb6 2084. 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 2506. (Revised by BJ, 22-Dec-2020.) |
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | sbt 2062 | A substitution into a theorem yields a theorem. See sbtALT 2065 for a shorter proof requiring more axioms. See chvar 2407 and chvarv 2408 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 2061. (Revised by Steven Nguyen, 6-Jul-2023.) |
⊢ 𝜑 ⇒ ⊢ [𝑡 / 𝑥]𝜑 | ||
Theorem | sbtru 2063 | The result of substituting in the truth constant "true" is true. (Contributed by BJ, 2-Sep-2023.) |
⊢ [𝑦 / 𝑥]⊤ | ||
Theorem | stdpc4 2064 | 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 3784 and rspsbc 3861. (Contributed by NM, 14-May-1993.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.) |
⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) | ||
Theorem | sbtALT 2065 | Alternate proof of sbt 2062, shorter but using ax-4 1801 and ax-5 1902. (Contributed by NM, 21-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 ⇒ ⊢ [𝑦 / 𝑥]𝜑 | ||
Theorem | 2stdpc4 2066 | A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2064 for the analogous single specialization. See 2sp 2175 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.) |
⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | sbi1 2067 | Distribute substitution over implication. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.) |
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | spsbim 2068 | Distribute substitution over implication. Closed form of sbimi 2070. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.) |
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)) | ||
Theorem | spsbbi 2069 | Biconditional property for substitution. Closed form of sbbii 2072. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)) | ||
Theorem | sbimi 2070 | Distribute substitution over implication. (Contributed by NM, 25-Jun-1998.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓) | ||
Theorem | sb2imi 2071 | Distribute substitution over implication. Compare al2imi 1807. (Contributed by Steven Nguyen, 13-Aug-2023.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒)) | ||
Theorem | sbbii 2072 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓) | ||
Theorem | 2sbbii 2073 | Infer double substitution into both sides of a logical equivalence. (Contributed by AV, 30-Jul-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓) | ||
Theorem | sbimdv 2074* | Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2236. (Contributed by Wolf Lammen, 6-May-2023.) Revise df-sb 2061. (Revised by Steven Nguyen, 6-Jul-2023.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒)) | ||
Theorem | sbbidv 2075* | Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 2237. (Contributed by Wolf Lammen, 6-May-2023.) (Proof shortened by Steven Nguyen, 6-Jul-2023.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑥]𝜒)) | ||
Theorem | sbbidvOLD 2076* | Obsolete version of sbbidv 2075 as of 6-Jul-2023. Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 2237. (Contributed by Wolf Lammen, 6-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) | ||
Theorem | sban 2077 | Conjunction inside and outside of a substitution are equivalent. Compare 19.26 1862. (Contributed by NM, 14-May-1993.) (Proof shortened by Steven Nguyen, 13-Aug-2023.) |
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) | ||
Theorem | sb3an 2078 | Threefold conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.) |
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓 ∧ [𝑦 / 𝑥]𝜒)) | ||
Theorem | spsbe 2079 | 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 2061. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.) |
⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) | ||
Theorem | spsbeOLD 2080 | Obsolete version of spsbe 2079 as of 11-Jul-2023. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) | ||
Theorem | sbequ 2081 | 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 2061. (Revised by BJ, 30-Dec-2020.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | ||
Theorem | sbequi 2082 | 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)) | ||
Theorem | sbequOLD 2083 | Obsolete proof of sbequ 2081 as of 7-Jul-2023. An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | ||
Theorem | sb6 2084* | 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 2500). Theorem sb6f 2533 replaces the disjoint variable condition with a non-freeness hypothesis. Theorem sb4b 2495 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2151. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.) |
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Theorem | 2sb6 2085* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) | ||
Theorem | sb1v 2086* | One direction of sb5 2268, provable from fewer axioms. Version of sb1 2499 with a disjoint variable condition using fewer axioms. (Contributed by Wolf Lammen, 20-Jan-2024.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | sb4vOLD 2087* | Obsolete as of 30-Jul-2023. Use sb6 2084 instead. (Contributed by BJ, 23-Jun-2019.) (Proof shortened by Steven Nguyen, 8-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb2vOLD 2088* | Obsolete as of 30-Jul-2023. Use sb6 2084 instead. Version of sb2 2500 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 31-May-2019.) Revise df-sb 2061. (Revised by Steven Nguyen, 8-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | ||
Theorem | sbv 2089* | Substitution for a variable not occurring in a proposition. See sbf 2262 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2089 can be proved directly by chaining equsv 2000 with sb6 2084. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | sbcom4 2090* | Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2091 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) | ||
Theorem | pm11.07 2091 | 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 2090 as earlier thought. See https://groups.google.com/g/metamath/c/iS0fOvSemC8/m/M1zTH8wxCAAJ 2090. (Contributed by BJ, 16-Sep-2018.) (New usage is discouraged.) |
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | sbrimvlem 2092* | Common proof template for sbrimvw 2093 and sbrimv 2306. The hypothesis is an instance of 19.21 2198. (Contributed by Wolf Lammen, 29-Jan-2024.) |
⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | sbrimvw 2093* | Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2305 and sbrimv 2306 based on fewer axioms, but with more disjoint variable conditions. Based on an idea of Gino Giotto. (Contributed by Wolf Lammen, 29-Jan-2024.) |
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | sbievw 2094* | Version of sbie 2540 and sbiev 2322 with more disjoint variable conditions, requiring fewer axioms. (Contributed by BJ, 18-Jul-2023.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbiedvw 2095* | Version of sbied 2541 and sbiedv 2542 with more disjoint variable conditions, requiring fewer axioms. (Contributed by Gino Giotto, 29-Jan-2024.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | 2sbievw 2096* | Version of 2sbiev 2543 with more disjoint variable conditions, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) |
⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓) | ||
Theorem | sbcom3vv 2097* | Version of sbcom3 2544 with a disjoint variable condition using fewer axioms. (Contributed by BJ, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 19-Jan-2023.) |
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑) | ||
Theorem | sbievw2 2098* | sbievw 2094 applied twice, avoiding a DV condition on 𝑥, 𝑦. Based on proofs by Wolf Lammen. (Contributed by Steven Nguyen, 29-Jul-2023.) |
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) & ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbco2vv 2099* | Version of sbco2 2549 with disjoint variable conditions and fewer axioms. (Contributed by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 29-Apr-2023.) |
⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | equsb3 2100* | Substitution in an equality. (Contributed by Raph Levien and FL, 4-Dec-2005.) Reduce axiom usage. (Revised by Wolf Lammen, 23-Jul-2023.) |
⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧) |
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