![]() |
Metamath
Proof Explorer Theorem List (p. 21 of 454) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28701) |
![]() (28702-30224) |
![]() (30225-45333) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | spimevw 2001* | Existential introduction, using implicit substitution. This is to spimew 1974 what spimvw 2002 is to spimw 1973. Version of spimev 2399 and spimefv 2196 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 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | spimvw 2002* | 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 2397 and spimfv 2239. (Contributed by NM, 9-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spvv 2003* | Specialization, using implicit substitution. Version of spv 2400 with a disjoint variable condition, which does not require ax-7 2015, ax-12 2175, ax-13 2379. (Contributed by NM, 30-Aug-1993.) (Revised by BJ, 31-May-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spfalw 2004 | Version of sp 2180 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 ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | chvarvv 2005* | Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvarv 2403 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 20-Apr-1994.) (Revised by BJ, 31-May-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | equs4v 2006* | Version of equs4 2427 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-May-1993.) (Revised by BJ, 31-May-2019.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | alequexv 2007* | Version of equs4v 2006 with its consequence simplified by exsimpr 1870. (Contributed by BJ, 9-Nov-2021.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
Theorem | exsbim 2008* | One direction of the equivalence in exsb 2367 is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.) |
⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
Theorem | equsv 2009* | 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 2090). (Contributed by BJ, 23-Jul-2023.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) | ||
Theorem | equsalvw 2010* | Version of equsalv 2265 with a disjoint variable condition, and of equsal 2428 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2011. (Contributed by BJ, 31-May-2019.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equsexvw 2011* | Version of equsexv 2266 with a disjoint variable condition, and of equsex 2429 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2010. (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 2012* | Obsolete version of equsexvw 2011 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 2013* | 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 2014* | 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 2015 |
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 2028). 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 2023 that this axiom can be recovered from its weakened version ax7v 2016 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 2015 should be ax7v 2016. See the comment of ax7v 2016 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 2023 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Theorem | ax7v 2016* |
Weakened version of ax-7 2015, with a disjoint variable condition on
𝑥,
𝑦. This should be
the only proof referencing ax-7 2015, and it
should be referenced only by its two weakened versions ax7v1 2017 and
ax7v2 2018, from which ax-7 2015
is then rederived as ax7 2023, which shows
that either ax7v 2016 or the conjunction of ax7v1 2017 and ax7v2 2018 is
sufficient.
In ax7v 2016, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 2016 "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 2021 and equid 2019 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 2023 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | ||
Theorem | ax7v1 2017* | First of two weakened versions of ax7v 2016, 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 2018* | Second of two weakened versions of ax7v 2016, 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 2019 | 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 2020 | 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 2021* | Weaker form of equcomi 2024 with a disjoint variable condition on 𝑥, 𝑦. This is an intermediate step and equcomi 2024 is fully recovered later. (Contributed by BJ, 7-Dec-2020.) |
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | ||
Theorem | ax6evr 2022* | A commuted form of ax6ev 1972. (Contributed by BJ, 7-Dec-2020.) |
⊢ ∃𝑥 𝑦 = 𝑥 | ||
Theorem | ax7 2023 |
Proof of ax-7 2015 from ax7v1 2017 and ax7v2 2018 (and earlier axioms), proving
sufficiency of the conjunction of the latter two weakened versions of
ax7v 2016, which is itself a weakened version of ax-7 2015.
Note that the weakened version of ax-7 2015 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 2024 | 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 2025 | Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.) |
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | ||
Theorem | equcomd 2026 | Deduction form of equcom 2025, symmetry of equality. For the versions for classes, see eqcom 2805 and eqcomd 2804. (Contributed by BJ, 6-Oct-2019.) |
⊢ (𝜑 → 𝑥 = 𝑦) ⇒ ⊢ (𝜑 → 𝑦 = 𝑥) | ||
Theorem | equcoms 2027 | 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 2028 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | ||
Theorem | equtrr 2029 | 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 2030 | Commuted version of equeucl 2031 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.) |
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥)) | ||
Theorem | equeucl 2031 | Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2015.) Curried (exported) form of equtr2 2034. (Contributed by BJ, 11-Apr-2021.) |
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | ||
Theorem | equequ1 2032 | 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 2033 | 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 2034 | Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2031. (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 2035 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 2249.) 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 2036* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2142, ax-13 2379. (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 2037* | A modified version of the forward implication of equvinv 2036 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.) |
⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)) | ||
Theorem | equvelv 2038* | A biconditional form of equvel 2468 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 2039 | An equivalence between two ways of expressing ax-13 2379. See the comment for ax-13 2379. (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 2040* | Weak version of sp 2180. 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 2041* | Weak version of the specialization scheme sp 2180. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2180 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2180 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2136 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2180 are spfw 2040 (minimal distinct variable requirements), spnfw 1984 (when 𝑥 is not free in ¬ 𝜑), spvw 1985 (when 𝑥 does not appear in 𝜑), sptruw 1808 (when 𝜑 is true), spfalw 2004 (when 𝜑 is false), and spvv 2003 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | cbvalw 2042* | 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 2043* | Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvalv 2407 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 2044* | Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvexv 2408 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 2045* | Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 2419 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.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | cbvexdvaw 2046* | Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 2420 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.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | cbval2vw 2047* | Rule used to change bound variables, using implicit substitution. Version of cbval2vv 2424 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
Theorem | cbvex2vw 2048* | Rule used to change bound variables, using implicit substitution. Version of cbvex2vv 2425 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | ||
Theorem | cbvex4vw 2049* | Rule used to change bound variables, using implicit substitution. Version of cbvex4v 2426 with more disjoint variable conditions, which requires fewer axioms. (Contributed by NM, 26-Jul-1995.) (Revised 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 2050* | Weak version of alcom 2160. 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 2051* | Obsolete version of alcomiw 2050 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 2052* | Weak version of ax-10 2142 from which we can prove any ax-10 2142 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 2053* | Weak version of hbn1 2143. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hba1w 2054* | Weak version of hba1 2297. See comments for ax10w 2130. 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 2055* | Weak version of hbe1 2144. See comments for ax10w 2130. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | ||
Theorem | hbalw 2056* | Weak version of hbal 2171. Uses only Tarski's FOL axiom schemes. Unlike hbal 2171, 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 2057* |
A special instance of sp 2180 applied to an equality with a disjoint
variable condition. Unlike the more general sp 2180, we
can prove this
without ax-12 2175. Instance of aeveq 2061.
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 2058* | Change bound variable in an equality with a disjoint variable condition. Instance of aev 2062. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦) | ||
Theorem | aevlem0 2059* | Lemma for aevlem 2060. Instance of aev 2062. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2175. (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 2060* | Lemma for aev 2062 and axc16g 2258. Change free and bound variables. Instance of aev 2062. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2379, 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 2061* | 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 2062* | A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2379, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2175. (Revised by Wolf Lammen, 19-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢) | ||
Theorem | aev2 2063* |
A version of aev 2062 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 2061, aev 2062,
aev2 2063).
Using aev 2062 and alrimiv 1928, 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 28245. This list cannot be extended to ¬ or ⊥ since the scheme ∀𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1797, ax-1 6-- ax-13 2379 (as the one-element universe shows). (Contributed by BJ, 23-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑡 𝑢 = 𝑣) | ||
Theorem | hbaev 2064* | All variables are effectively bound in an identical variable specifier. Version of hbae 2442 with a disjoint variable condition, requiring fewer axioms. Instance of aev2 2063. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 22-Mar-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) | ||
Theorem | naev 2065* | 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 2066* | Generalization of hbnaev 2067. (Contributed by Wolf Lammen, 9-Apr-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑡 𝑡 = 𝑢) | ||
Theorem | hbnaev 2067* | Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. This condition is dropped in hbnae 2443, at the expense of more axiom dependencies. Instance of naev2 2066. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 9-Apr-2021.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | sbjust 2068* | Justification theorem for df-sb 2070 proved from Tarski's FOL axiom schemes. (Contributed by BJ, 22-Jan-2023.) |
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | ||
Syntax | wsb 2069 | 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 2070* |
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 2127.
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 2088, sbcom2 2165 and sbid2v 2528). Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2254 shows. We achieve this by applying twice Tarski's definition sb6 2090 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2281 with respect to sb5 2273. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2499 shows. Another version that mixes free and bound variables is dfsb3 2512. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2273 and sb6 2090. 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 2499. (Revised by BJ, 22-Dec-2020.) |
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | sbt 2071 | A substitution into a theorem yields a theorem. See sbtALT 2074 for a shorter proof requiring more axioms. See chvar 2402 and chvarv 2403 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 2070. (Revised by Steven Nguyen, 6-Jul-2023.) |
⊢ 𝜑 ⇒ ⊢ [𝑡 / 𝑥]𝜑 | ||
Theorem | sbtru 2072 | The result of substituting in the truth constant "true" is true. (Contributed by BJ, 2-Sep-2023.) |
⊢ [𝑦 / 𝑥]⊤ | ||
Theorem | stdpc4 2073 | 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 3733 and rspsbc 3808. (Contributed by NM, 14-May-1993.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) |
⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑) | ||
Theorem | sbtALT 2074 | Alternate proof of sbt 2071, shorter but using ax-4 1811 and ax-5 1911. (Contributed by NM, 21-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 ⇒ ⊢ [𝑦 / 𝑥]𝜑 | ||
Theorem | 2stdpc4 2075 | A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2073 for the analogous single specialization. See 2sp 2183 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 2076 | 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 2077 | Distribute substitution over implication. Closed form of sbimi 2079. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2070. (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 2078 | Biconditional property for substitution. Closed form of sbbii 2081. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)) | ||
Theorem | sbimi 2079 | Distribute substitution over implication. (Contributed by NM, 25-Jun-1998.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓) | ||
Theorem | sb2imi 2080 | Distribute substitution over implication. Compare al2imi 1817. (Contributed by Steven Nguyen, 13-Aug-2023.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒)) | ||
Theorem | sbbii 2081 | 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 2082 | 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 2083* | Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2243. (Contributed by Wolf Lammen, 6-May-2023.) Revise df-sb 2070. (Revised by Steven Nguyen, 6-Jul-2023.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒)) | ||
Theorem | sbbidv 2084* | Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 2244. (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 | sban 2085 | Conjunction inside and outside of a substitution are equivalent. Compare 19.26 1871. (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 2086 | 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 2087 | 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 2070. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.) |
⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) | ||
Theorem | sbequ 2088 | 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 2070. (Revised by BJ, 30-Dec-2020.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | ||
Theorem | sbequi 2089 | 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 | sb6 2090* | 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 2493). Theorem sb6f 2515 replaces the disjoint variable condition with a non-freeness hypothesis. Theorem sb4b 2488 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2158. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.) |
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Theorem | 2sb6 2091* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) | ||
Theorem | sb1v 2092* | One direction of sb5 2273, provable from fewer axioms. Version of sb1 2492 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 = 𝑦 ∧ 𝜑)) | ||
Theorem | sb4vOLD 2093* | Obsolete as of 30-Jul-2023. Use sb6 2090 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 2094* | Obsolete as of 30-Jul-2023. Use sb6 2090 instead. Version of sb2 2493 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by BJ, 31-May-2019.) Revise df-sb 2070. (Revised by Steven Nguyen, 8-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | ||
Theorem | sbv 2095* | Substitution for a variable not occurring in a proposition. See sbf 2268 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2095 can be proved directly by chaining equsv 2009 with sb6 2090. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | sbcom4 2096* | Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2097 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 2097 | 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 2096 as earlier thought. See https://groups.google.com/g/metamath/c/iS0fOvSemC8/m/M1zTH8wxCAAJ 2096. (Contributed by BJ, 16-Sep-2018.) (New usage is discouraged.) |
⊢ 𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | sbrimvlem 2098* | Common proof template for sbrimvw 2099 and sbrimv 2310. The hypothesis is an instance of 19.21 2205. (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 2099* | Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2309 and sbrimv 2310 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 2100* | Conversion of implicit substitution to explicit substitution. Version of sbie 2521 and sbiev 2322 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |