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Theorem List for Metamath Proof Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfsb2 2501 Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theoremhbsb2a 2502 Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremsb4e 2503 One direction of a simplified definition of substitution that unlike sb4b 2488 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremhbsb2e 2504 Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Theoremhbsb3 2505 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out bj-hbsb3v 34319 for a weaker version requiring less axioms. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑦𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremnfs1 2506 If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out nfs1v 2157 for a version requiring less axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
𝑦𝜑       𝑥[𝑦 / 𝑥]𝜑

Theoremaxc16ALT 2507* Alternate proof of axc16 2259, shorter but requiring ax-10 2142, ax-11 2158, ax-13 2379 and using df-nf 1786 and df-sb 2070. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremaxc16gALT 2508* Alternate proof of axc16g 2258 that uses df-sb 2070 and requires ax-10 2142, ax-11 2158, ax-13 2379. (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 2509 Substitution applied to an atomic wff. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker equsb1v 2109 if possible. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
[𝑦 / 𝑥]𝑥 = 𝑦

Theoremequsb2 2510 Substitution applied to an atomic wff. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out equsb1v 2109 for a version requiring less axioms. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
[𝑦 / 𝑥]𝑦 = 𝑥

Theoremdfsb2 2511 An alternate definition of proper substitution that, like dfsb1 2499, mixes free and bound variables to avoid distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 17-Feb-2005.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremdfsb3 2512 An alternate definition of proper substitution df-sb 2070 that uses only primitive connectives (no defined terms) on the right-hand side. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 6-Mar-2007.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremdrsb1 2513 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Jun-1993.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))

Theoremsb2ae 2514* In the case of two successive substitutions for two always equal variables, the second substitution has no effect. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by BJ and WL, 9-Aug-2023.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))

Theoremsb6f 2515 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2493 and does not require the non-freeness hypothesis. Theorem sb6 2090 replaces the non-freeness hypothesis with a disjoint variable condition and uses less axioms. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
𝑦𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5f 2516 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2492 and does not require the non-freeness hypothesis. Theorem sb5 2273 replaces the non-freeness hypothesis with a disjoint variable condition and uses less axioms. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
𝑦𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremnfsb4t 2517 A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 2518). Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)
(∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))

Theoremnfsb4 2518 A variable not free in a proposition remains so after substitution in that proposition with a distinct variable. Usage of this theorem is discouraged because it depends on ax-13 2379. Theorem nfsb 2542 replaces the distinctor with a disjoint variable condition. Visit also nfsbv 2338 for a weaker version of nfsb 2542 not requiring ax-13 2379. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
𝑧𝜑       (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Theoremsbi1OLD 2519 Obsolete version of sbi1 2076 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 2520 Elimination of equality from antecedent after substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.) Revise df-sb 2070. (Revised by Wolf Lammen, 28-Jul-2023.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))

Theoremsbie 2521 Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev 2322 and sbievw 2100. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝑦 / 𝑥]𝜑𝜓)

Theoremsbied 2522 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2521) Usage of this theorem is discouraged because it depends on ax-13 2379. See sbiedw 2323, sbiedvw 2101 for variants using disjoint variables, but requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theoremsbiedv 2523* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2521). Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker sbiedvw 2101 when possible. (Contributed by NM, 7-Jan-2017.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))

Theorem2sbiev 2524* Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. See 2sbievw 2102 for a version with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023.) (New usage is discouraged.)
((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)

Theoremsbcom3 2525 Substituting 𝑦 for 𝑥 and then 𝑧 for 𝑦 is equivalent to substituting 𝑧 for both 𝑥 and 𝑦. Usage of this theorem is discouraged because it depends on ax-13 2379. For a version requiring a disjoint variable, but fewer axioms, see sbcom3vv 2103. (Contributed by Giovanni Mascellani, 8-Apr-2018.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 16-Sep-2018.) (Proof shortened by Wolf Lammen, 16-Sep-2018.) (New usage is discouraged.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)

Theoremsbco 2526 A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. See sbcov 2255 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.) (New usage is discouraged.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbid2 2527 An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out sbid2vw 2257 for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑥𝜑       ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbid2v 2528* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). Usage of this theorem is discouraged because it depends on ax-13 2379. See sbid2vw 2257 for a version with an extra disjoint variable condition requiring fewer axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑𝜑)

Theoremsbidm 2529 An idempotent law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.)
([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco2 2530 A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v 2341 and sbco2vv 2105. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (New usage is discouraged.)
𝑧𝜑       ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsbco2d 2531 A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Theoremsbco3 2532 A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.) (New usage is discouraged.)
([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Theoremsbcom 2533 A commutativity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out sbcom3vv 2103 for a version requiring less axioms. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) (New usage is discouraged.)
([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)

Theoremsbtrt 2534 Partially closed form of sbtr 2535. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by BJ, 4-Jun-2019.) (New usage is discouraged.)
𝑦𝜑       (∀𝑦[𝑦 / 𝑥]𝜑𝜑)

Theoremsbtr 2535 A partial converse to sbt 2071. If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by BJ, 15-Sep-2018.) (New usage is discouraged.)
𝑦𝜑    &   [𝑦 / 𝑥]𝜑       𝜑

Theoremsb8 2536 Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 2379. For a version requiring disjoint variables, but fewer axioms, see sb8v 2362. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)
𝑦𝜑       (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb8e 2537 Substitution of variable in existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2379. For a version requiring disjoint variables, but fewer axioms, see sb8ev 2363. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)
𝑦𝜑       (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Theoremsb9 2538 Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2539. (Revised by Wolf Lammen, 15-Jun-2019.) (New usage is discouraged.)
(∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsb9i 2539 Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Jun-2019.) (New usage is discouraged.)
(∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

Theoremsbhb 2540* Two ways of expressing "𝑥 is (effectively) not free in 𝜑". Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 29-May-2009.) (New usage is discouraged.)
((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))

Theoremnfsbd 2541* Deduction version of nfsb 2542. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 15-Feb-2013.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑧𝜓)       (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)

Theoremnfsb 2542* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Usage of this theorem is discouraged because it depends on ax-13 2379. For a version requiring more disjoint variables, but fewer axioms, see nfsbv 2338. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Feb-2024.) (New usage is discouraged.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑

TheoremnfsbOLD 2543* Obsolete version of nfsb 2542 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 2544* If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker hbsbw 2173 when possible. (Contributed by NM, 12-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Theoremsb7f 2545* This version of dfsb7 2281 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 1911 i.e. that doesn't have the concept of a variable not occurring in a wff. (dfsb1 2499 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑧𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremsb7h 2546* This version of dfsb7 2281 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 1911 i.e. that doesn't have the concept of a variable not occurring in a wff. (dfsb1 2499 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremsb10f 2547* 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. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑥𝜑       ([𝑦 / 𝑧]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑧]𝜑))

Theoremsbal1 2548* Check out sbal 2163 for a version not dependent on ax-13 2379. A theorem used in elimination of disjoint variable restriction on 𝑥 and 𝑧 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 2549* Move quantifier in and out of substitution. Usage of this theorem is discouraged because it depends on ax-13 2379. Check out sbal 2163 for a version replacing the distinctor with a disjoint variable condition, requiring fewer axioms. (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.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Theoremsbal2OLD 2550* Obsolete version of sbal2 2549 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 2551* Obsolete version of sbal 2163 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 2552* An equivalent expression for double existence. Usage of this theorem is discouraged because it depends on ax-13 2379. For a version requiring more disjoint variables, but fewer axioms, see 2sb8ev 2364. (Contributed by Wolf Lammen, 2-Nov-2019.) (New usage is discouraged.)
(∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

1.5.5  Alternate definition of substitution

The definition of substitution (df-sb 2070) used to be dfsb1 2499. These two definitions are proved equivalent by proving dfsb7 2281 from both, which takes several intermediate theorems and uses many axioms.

TheoremsbimiALT 2553 Alternate version of sbimi 2079. (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 2554 Alternate version of sbbii 2081. (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 2555 Alternate version of sbequ1 2246. (Contributed by NM, 16-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝑥 = 𝑦 → (𝜑𝜃))

Theoremsbequ2ALT 2556 Alternate version of sbequ2 2247. (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 2557 Alternate version of sbequ12 2250. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝑥 = 𝑦 → (𝜑𝜃))

Theoremsb1ALT 2558 Alternate version of sb1 2492. (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 2559* Alternate version of sb2vOLD 2094. (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 2560* Alternate version of sb4vOLD 2093. (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 2561* Alternate version of sb6 2090. (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 2562* Alternate version of sb5 2273. (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 2563 Alternate version of sb2 2493. (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 2564 Alternate version of one implication of sb4b 2488. (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 2565 Alternate version of spsbe 2087. (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 2566 Alternate version of stdpc4 2073. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (∀𝑥𝜑𝜃)

Theoremdfsb2ALT 2567 Alternate version of dfsb2 2511. (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 2568 Alternate version of dfsb3 2512. (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 2569 Alternate version of sbft 2267. (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 2570 Alternate version of sbf 2268. (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 2571 Alternate version of sbn 2283. (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 2572 Alternate version of sbequi 2089. (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 2573 Alternate version of sbequ 2088. (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 2574 Alternate version of sb4a 2498. (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 2575 Alternate version of hbsb2 2500. (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 2576 Alternate version of nfsb2 2501. (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 2577 Alternate version of equsb1 2509. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦)))       𝜃

Theoremsb6fALT 2578 Alternate version of sb6f 2515. (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 2579 Alternate version of sb5f 2516. (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 2580 Alternate version of nfsb4t 2517. (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))

Theoremnfsb4ALT 2581 Alternate version of nfsb4 2518. (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)))    &   𝑧𝜑       (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃)

Theoremsbi1ALT 2582 Alternate version of sbi1 2076. (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)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 → (𝜃𝜏))

Theoremsbi2ALT 2583 Alternate version of sbi2 2306. (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)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       ((𝜃𝜏) → 𝜂)

TheoremsbimALT 2584 Alternate version of sbim 2307. (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)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 ↔ (𝜃𝜏))

TheoremsbrimALT 2585 Alternate version of sbrim 2309. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))    &   𝑥𝜑       (𝜂 ↔ (𝜑𝜏))

TheoremsbanALT 2586 Alternate version of sban 2085. (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)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 ↔ (𝜃𝜏))

TheoremsbbiALT 2587 Alternate version of sbbi 2313. (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)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))       (𝜂 ↔ (𝜃𝜏))

TheoremsblbisALT 2588 Alternate version of sblbis 2314. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))    &   (𝜏𝜒)       (𝜂 ↔ (𝜃𝜒))

TheoremsbieALT 2589 Alternate version of sbie 2521. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜃𝜓)

TheoremsbiedALT 2590 Alternate version of sbied 2522. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))    &   𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (𝜏𝜒))

Theoremsbco2ALT 2591 Alternate version of sbco2 2530. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   (𝜏 ↔ ((𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑))) ∧ ∃𝑧(𝑧 = 𝑦 ∧ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))))    &   𝑧𝜑       (𝜏𝜃)

Theoremsb7fALT 2592* Alternate version of sb7f 2545. (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))    &   𝑧𝜑       (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

Theoremdfsb7ALT 2593* Alternate version of dfsb7 2281. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))       (𝜃 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧𝜑)))

1.6  Uniqueness and unique existence

Theoremdfmoeu 2594* An elementary proof of moeu 2643 in disguise, connecting an expression characterizing uniqueness (df-mo 2598) to that of existential uniqueness (eu6 2634). No particular order of definition is required, as one can be derived from the other. This is shown here and in dfeumo 2595. (Contributed by Wolf Lammen, 27-May-2019.)
((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremdfeumo 2595* An elementary proof showing the reverse direction of dfmoeu 2594. Here the characterizing expression of existential uniqueness (eu6 2634) is derived from that of uniqueness (df-mo 2598). (Contributed by Wolf Lammen, 3-Oct-2023.)
((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

1.6.1  Uniqueness: the at-most-one quantifier

Syntaxwmo 2596 Extend wff definition to include the at-most-one quantifier ("there exists at most one 𝑥 such that 𝜑").
wff ∃*𝑥𝜑

Theoremmojust 2597* Soundness justification theorem for df-mo 2598 (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010.) Added this theorem by adapting the proof of eujust 2631. (Revised by BJ, 30-Sep-2022.)
(∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))

Definitiondf-mo 2598* Define the at-most-one quantifier. The expression ∃*𝑥𝜑 is read "there exists at most one 𝑥 such that 𝜑". This is also called the "uniqueness quantifier" but that expression is also used for the unique existential quantifier df-eu 2629, therefore we avoid that ambiguous name.

Notation of [BellMachover] p. 460, whose definition we show as mo3 2623. For other possible definitions see moeu 2643 and mo4 2625. (Contributed by Wolf Lammen, 27-May-2019.) Make this the definition (which used to be moeu 2643, while this definition was then proved as dfmo 2657). (Revised by BJ, 30-Sep-2022.)

(∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))

Theoremnexmo 2599 Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2158. (Revised by Wolf Lammen, 16-Oct-2022.)
(¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Theoremexmo 2600 Any proposition holds for some 𝑥 or holds for at most one 𝑥. (Contributed by NM, 8-Mar-1995.) Shorten proof and avoid df-eu 2629. (Revised by BJ, 14-Oct-2022.)
(∃𝑥𝜑 ∨ ∃*𝑥𝜑)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45415
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