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Theorem List for Metamath Proof Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsb3OLD 2501 Obsolete version of sb3 2498 as of 21-Feb-2024. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
 
Theoremsb4OLD 2502 Obsolete as of 30-Jul-2023. Use sb4b 2495 instead. One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993.) Revise df-sb 2061. (Revised by Wolf Lammen, 25-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremsb1OLD 2503 Obsolete version of sb1 2499 as of 21-Feb-2024. (Contributed by NM, 13-May-1993.) Revise df-sb 2061. (Revised by Wolf Lammen, 29-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb3bOLD 2504 Obsolete version of sb3b 2497 as of 21-Feb-2024. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
 
Theoremsb4a 2505 A version of one implication of sb4b 2495 that does not require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) Revise df-sb 2061. (Revised by Wolf Lammen, 28-Jul-2023.)
([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))
 
Theoremdfsb1 2506 Alternate definition of substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). This was the original definition before df-sb 2061. Note that it does not require dummy variables in its definiens; this is done by having 𝑥 free in the first conjunct and bound in the second. (Contributed by BJ, 9-Jul-2023.) Revise df-sb 2061. (Revised by Wolf Lammen, 29-Jul-2023.)
([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
 
TheoremspsbeOLDOLD 2507 Obsolete version of spsbe 2079 as of 7-Jul-2023. A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
 
Theoremsb2vOLDOLD 2508* Obsolete version of sb2 2500 as of 8-Jul-2023. Version of sb2 2500 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by BJ, 31-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
 
Theoremsb4vOLDOLD 2509* Obsolete version of sb4vOLD 2087 as of 8-Jul-2023. Version of sb4OLD 2502 with a disjoint variable condition instead of a distinctor antecedent, which does not require ax-13 2383. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremsbequ2OLDOLD 2510 Obsolete version of sbequ2 2241 as of 8-Jul-2023. An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
 
TheoremsbimiOLD 2511 Obsolete version of sbimi 2070 as of 6-Jul-2023. Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
 
TheoremsbimdvOLD 2512* Obsolete version of sbimdv 2074 as of 6-Jul-2023. Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2236. (Contributed by Wolf Lammen, 6-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
 
Theoremequsb1vOLDOLD 2513* Obsolete version of equsb1v 2103 as of 19-Jun-2023. (Contributed by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 31-May-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
[𝑦 / 𝑥]𝑥 = 𝑦
 
TheoremsbimdOLD 2514 Obsolete version of sbimd as of 9-Jul-2023. (Contributed by Wolf Lammen, 24-Nov-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
 
TheoremsbtvOLD 2515* Obsolete version of sbt 2062 as of 6-Jul-2023. A substitution into a theorem yields a theorem. See sbt 2062 when 𝑥, 𝑦 need not be disjoint. (Contributed by BJ, 31-May-2019.) Reduce axioms. (Revised by Steven Nguyen, 25-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑       [𝑥 / 𝑦]𝜑
 
Theoremsbequ1OLD 2516 Obsolete version of sbequ1 2240 as of 8-Jul-2023. An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
 
Theoremhbsb2 2517 Bound-variable hypothesis builder for substitution. (Contributed by NM, 14-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
 
Theoremnfsb2 2518 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
 
Theoremhbsb2a 2519 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
 
Theoremsb4e 2520 One direction of a simplified definition of substitution that unlike sb4b 2495 does not require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 
Theoremhbsb2e 2521 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
 
Theoremhbsb3 2522 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 14-May-1993.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
 
Theoremnfs1 2523 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑦𝜑       𝑥[𝑦 / 𝑥]𝜑
 
Theoremaxc16ALT 2524* Alternate proof of axc16 2253, shorter but requiring ax-10 2136, ax-11 2151, ax-13 2383 and using df-nf 1776 and df-sb 2061. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
Theoremaxc16gALT 2525* Alternate proof of axc16g 2252 that uses df-sb 2061 and requires ax-10 2136, ax-11 2151, ax-13 2383. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 
Theoremequsb1 2526 Substitution applied to an atomic wff. (Contributed by NM, 10-May-1993.)
[𝑦 / 𝑥]𝑥 = 𝑦
 
Theoremequsb2 2527 Substitution applied to an atomic wff. (Contributed by NM, 10-May-1993.)
[𝑦 / 𝑥]𝑦 = 𝑥
 
Theoremdfsb2 2528 An alternate definition of proper substitution that, like dfsb1 2506, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)
([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremdfsb3 2529 An alternate definition of proper substitution df-sb 2061 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
 
TheoremsbequiOLD 2530 Obsolete proof of sbequi 2082 as of 7-Jul-2023. An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
 
Theoremdrsb1 2531 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 2-Jun-1993.)
(∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
 
Theoremsb2ae 2532* In the case of two successive substitutions for two always equal variables, the second substitution has no effect. (Contributed by BJ and WL, 9-Aug-2023.)
(∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))
 
Theoremsb6f 2533 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2500 and does not require the non-freeness hypothesis. Theorem sb6 2084 replaces the non-freeness hypothesis with a disjoint variable condition. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑦𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb5f 2534 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2499 and does not require the non-freeness hypothesis. Theorem sb5 2268 replaces the non-freeness hypothesis with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑦𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremnfsb4t 2535 A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 2536). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
(∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
 
Theoremnfsb4 2536 A variable not free in a proposition remains so after substitution in that proposition with a distinct variable. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑧𝜑       (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
 
TheoremsbnOLD 2537 Obsolete version of sbn 2279 as of 8-Jul-2023. Negation inside and outside of substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbnvOLD 2314. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
 
Theoremsbi1OLD 2538 Obsolete version of sbi1 2067 as of 24-Jul-2023. Removal of implication from substitution. (Contributed by NM, 14-May-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 
Theoremsbequ8 2539 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.) Revise df-sb 2061. (Revised by Wolf Lammen, 28-Jul-2023.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
 
Theoremsbie 2540 Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev 2322 and sbievw 2094. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)
 
Theoremsbied 2541 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2540) See sbiedv 2542, sbiedw 2324, sbiedvw 2095 for variants using disjoint variables, but require fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
Theoremsbiedv 2542* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2540). See sbied 2541, sbiedvw 2095, sbiedw 2324 for similar variants (Contributed by NM, 7-Jan-2017.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
Theorem2sbiev 2543* Conversion of double implicit substitution to explicit substitution. See 2sbievw 2096 for a variant with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023.)
((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
 
Theoremsbcom3 2544 Substituting 𝑦 for 𝑥 and then 𝑧 for 𝑦 is equivalent to substituting 𝑧 for both 𝑥 and 𝑦. For a version requiring disjoint variables, but fewer axioms, see sbcom3vv 2097. (Contributed by Giovanni Mascellani, 8-Apr-2018.) Remove dependency on ax-11 2151. (Revised by Wolf Lammen, 16-Sep-2018.) (Proof shortened by Wolf Lammen, 16-Sep-2018.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
 
Theoremsbco 2545 A composition law for substitution. See sbcov 2249 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theoremsbid2 2546 An identity law for substitution. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
 
Theoremsbid2v 2547* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). See sbid2vw 2251 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 5-Aug-1993.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)
 
Theoremsbidm 2548 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.)
([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theoremsbco2 2549 A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v 2344 and sbco2vv 2099. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.)
𝑧𝜑       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
Theoremsbco2d 2550 A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
 
Theoremsbco3 2551 A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
 
Theoremsbcom 2552 A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-2018.)
([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
 
Theoremsbtrt 2553 Partially closed form of sbtr 2554. (Contributed by BJ, 4-Jun-2019.)
𝑦𝜑       (∀𝑦[𝑦 / 𝑥]𝜑𝜑)
 
Theoremsbtr 2554 A partial converse to sbt 2062. If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. (Contributed by BJ, 15-Sep-2018.)
𝑦𝜑    &   [𝑦 / 𝑥]𝜑       𝜑
 
Theoremsb8 2555 Substitution of variable in universal quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8v 2365. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb8e 2556 Substitution of variable in existential quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2366. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
𝑦𝜑       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb9 2557 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2558. (Revised by Wolf Lammen, 15-Jun-2019.)
(∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsb9i 2558 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Jun-2019.)
(∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)
 
Theoremsbhb 2559* Two ways of expressing "𝑥 is (effectively) not free in 𝜑". (Contributed by NM, 29-May-2009.)
((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
 
Theoremnfsbd 2560* Deduction version of nfsb 2561. (Contributed by NM, 15-Feb-2013.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
 
Theoremnfsb 2561* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. For a version requiring more disjoint variables, but fewer axioms, see nfsbv 2341. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Feb-2024.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑
 
TheoremnfsbOLD 2562* Obsolete version of nfsb 2561 as of 25-Feb-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑
 
Theoremhbsb 2563* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 
Theoremsb7f 2564* This version of dfsb7 2277 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1902 i.e. that doesn't have the concept of a variable not occurring in a wff. (dfsb1 2506 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑧𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
 
Theoremsb7h 2565* This version of dfsb7 2277 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1902 i.e. that doesn't have the concept of a variable not occurring in a wff. (dfsb1 2506 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
 
Theoremdfsb7OLDOLD 2566* Obsolete version of dfsb7 2277 as of 8-Jul-2023.

An alternate definition of proper substitution df-sb 2061. By introducing a dummy variable 𝑧 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑥, 𝑦, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑧 effectively insulates 𝑥 from 𝑦. To achieve this, we use a chain of two substitutions in the form of sb5 2268, first 𝑧 for 𝑥 then 𝑦 for 𝑧. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2800. Theorem sb7h 2565 provides a version where 𝜑 and 𝑧 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
 
Theoremsb10f 2567* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))
 
Theoremsbal1 2568* Obsolete version of sbal 2156 as of 13-Jul-2023. A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑦 by replacing it with a distinctor ¬ ∀𝑥𝑥 = 𝑧. (Contributed by NM, 15-May-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremsbal2 2569* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 24-Dec-2022.) (Proof shortened by Wolf Lammen, 23-Sep-2023.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
Theoremsbal2OLD 2570* Obsolete version of sbal2 2569 as of 23-Sep-2023. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 24-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
TheoremsbalOLD 2571* Obsolete version of sbal 2156 as of 13-Aug-2023. Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
 
Theorem2sb8e 2572* An equivalent expression for double existence. For a version requiring more disjoint variables, but fewer axioms, see 2sb8ev 2367. (Contributed by Wolf Lammen, 2-Nov-2019.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
 
1.5.5  Alternate definition of substitution

The definition of substitution (df-sb 2061) used to be dfsb1 2506. These two definitions are proved equivalent by proving dfsb7 2277 from both, which takes several intermediate theorems and uses many axioms.

 
TheoremsbimiALT 2573 Alternate version of sbimi 2070. (Contributed by NM, 25-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜑𝜓)       (𝜃𝜏)
 
TheoremsbbiiALT 2574 Alternate version of sbbii 2072. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜑𝜓)       (𝜃𝜏)
 
Theoremsbequ1ALT 2575 Alternate version of sbequ1 2240. (Contributed by NM, 16-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝑥 = 𝑦 → (𝜑𝜃))
 
Theoremsbequ2ALT 2576 Alternate version of sbequ2 2241. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝑥 = 𝑦 → (𝜃𝜑))
 
Theoremsbequ12ALT 2577 Alternate version of sbequ12 2244. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝑥 = 𝑦 → (𝜑𝜃))
 
Theoremsb1ALT 2578 Alternate version of sb1 2499. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 → ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb2vOLDALT 2579* Alternate version of sb2vOLD 2088. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (∀𝑥(𝑥 = 𝑦𝜑) → 𝜃)
 
Theoremsb4vOLDALT 2580* Alternate version of sb4vOLD 2087. (Contributed by BJ, 23-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 → ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb6ALT 2581* Alternate version of sb6 2084. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb5ALT2 2582* Alternate version of sb5 2268. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb2ALT 2583 Alternate version of sb2 2500. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (∀𝑥(𝑥 = 𝑦𝜑) → 𝜃)
 
Theoremsb4ALT 2584 Alternate version of one implication of sb4b 2495. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
TheoremspsbeALT 2585 Alternate version of spsbe 2079. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 → ∃𝑥𝜑)
 
Theoremstdpc4ALT 2586 Alternate version of stdpc4 2064. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (∀𝑥𝜑𝜃)
 
Theoremdfsb2ALT 2587 Alternate version of dfsb2 2528. (Contributed by NM, 17-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremdfsb3ALT 2588 Alternate version of dfsb3 2529. (Contributed by NM, 6-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
 
TheoremsbftALT 2589 Alternate version of sbft 2261. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (Ⅎ𝑥𝜑 → (𝜃𝜑))
 
TheoremsbfALT 2590 Alternate version of sbf 2262. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑥𝜑       (𝜃𝜑)
 
TheoremsbnALT 2591 Alternate version of sbn 2279. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)))       (𝜏 ↔ ¬ 𝜃)
 
TheoremsbequiALT 2592 Alternate version of sbequi 2082. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑧 = 𝑥𝜑) ∧ ∃𝑧(𝑧 = 𝑥𝜑)))    &   (𝜏 ↔ ((𝑧 = 𝑦𝜑) ∧ ∃𝑧(𝑧 = 𝑦𝜑)))       (𝑥 = 𝑦 → (𝜃𝜏))
 
TheoremsbequALT 2593 Alternate version of sbequ 2081. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑧 = 𝑥𝜑) ∧ ∃𝑧(𝑧 = 𝑥𝜑)))    &   (𝜏 ↔ ((𝑧 = 𝑦𝜑) ∧ ∃𝑧(𝑧 = 𝑦𝜑)))       (𝑥 = 𝑦 → (𝜃𝜏))
 
Theoremsb4aALT 2594 Alternate version of sb4a 2505. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦 → ∀𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)))       (𝜃 → ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremhbsb2ALT 2595 Alternate version of hbsb2 2517. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥𝜃))
 
Theoremnfsb2ALT 2596 Alternate version of nfsb2 2518. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜃)
 
Theoremequsb1ALT 2597 Alternate version of equsb1 2526. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦)))       𝜃
 
Theoremsb6fALT 2598 Alternate version of sb6f 2533. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑦𝜑       (𝜃 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb5fALT 2599 Alternate version of sb5f 2534. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑦𝜑       (𝜃 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremnfsb4tALT 2600 Alternate version of nfsb4t 2535. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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