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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxc15 2301 Derivation of set.mm's original ax-c15 33993 from ax-c11n 33992 and the shorter ax-12 2045 that has replaced it.

Theorem ax12 2302 shows the reverse derivation of ax-12 2045 from ax-c15 33993.

Normally, axc15 2301 should be used rather than ax-c15 33993, except by theorems specifically studying the latter's properties. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12 2302 Rederivation of axiom ax-12 2045 from ax12v 2046 (used only via sp 2051) , axc11r 2185, and axc15 2301 (on top of Tarski's FOL). (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13ALT 2303 Alternate proof of ax13 2247 from FOL, sp 2051, and axc9 2300. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremaxc11nlemALT 2304* Alternate version of axc11nlemOLD2 1986 used in an older proof. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑤 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11n 2305 Derive set.mm's original ax-c11n 33992 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). If a disjoint variable condition is added on 𝑥 and 𝑦, then this becomes an instance of aevlem 1979. Use aecom 2309 instead when this does not lengthen the proof. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 2-Jul-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11nOLD 2306 Obsolete proof of axc11n 2305 as of 2-Jul-2021. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11nOLDOLD 2307 Old proof of axc11n 2305. Obsolete as of 29-Mar-2021. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Adapt to a modification of axc11nlemOLD2 1986. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11nALT 2308 Alternate proof of axc11n 2305 from axc11nlemALT 2304. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecom 2309 Commutation law for identical variable specifiers. Both sides of the biconditional are true when 𝑥 and 𝑦 are substituted with the same variable. (Contributed by NM, 10-May-1993.) Changed to a biconditional. (Revised by BJ, 26-Sep-2019.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)

Theoremaecoms 2310 A commutation rule for identical variable specifiers. (Contributed by NM, 10-May-1993.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremnaecoms 2311 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremaxc11 2312 Show that ax-c11 33991 can be derived from ax-c11n 33992 in the form of axc11n 2305. Normally, axc11 2312 should be used rather than ax-c11 33991, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremhbae 2313 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremnfae 2314 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧𝑥 𝑥 = 𝑦

Theoremhbnae 2315 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 13-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremnfnae 2316 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦

Theoremhbnaes 2317 Rule that applies hbnae 2315 to antecedent. (Contributed by NM, 15-May-1993.)
(∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑥 𝑥 = 𝑦𝜑)

TheoremaevlemALTOLD 2318* Older alternate version of aevlem 1979. Obsolete as of 30-Mar-2021. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥)

TheoremaevALTOLD 2319* Older alternate proof of aev 1981. Obsolete as of 30-Mar-2021. (Contributed by NM, 8-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Theoremaxc16i 2320* Inference with axc16 2133 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremaxc16nfALT 2321* Alternate proof of axc16nf 2135, shorter but requiring ax-11 2032 and ax-13 2244. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theoremdral2 2322 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, 27-Feb-2005.) Allow a shortening of dral1 2323. (Revised by Wolf Lammen, 4-Mar-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremdral1 2323 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, 24-Nov-1994.) Remove dependency on ax-11 2032. (Revised by Wolf Lammen, 6-Sep-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremdral1ALT 2324 Alternate proof of dral1 2323, shorter but requiring ax-11 2032. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremdrex1 2325 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, 27-Feb-2005.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Theoremdrex2 2326 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, 27-Feb-2005.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 ↔ ∃𝑧𝜓))

Theoremdrnf1 2327 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Theoremdrnf2 2328 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Theoremnfald2 2329 Variation on nfald 2163 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremnfexd2 2330 Variation on nfexd 2165 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremexdistrf 2331 Distribution of existential quantifiers, with a bound-variable hypothesis saying that 𝑦 is not free in 𝜑, but 𝑥 can be free in 𝜑 (and there is no distinct variable condition on 𝑥 and 𝑦). (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)       (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theoremdvelimf 2332 Version of dvelimv 2336 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
𝑥𝜑    &   𝑧𝜓    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Theoremdvelimdf 2333 Deduction form of dvelimf 2332. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑧𝜒)    &   (𝜑 → (𝑧 = 𝑦 → (𝜓𝜒)))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Theoremdvelimh 2334 Version of dvelim 2335 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelim 2335* This theorem can be used to eliminate a distinct variable restriction on 𝑥 and 𝑧 and replace it with the "distinctor" ¬ ∀𝑥𝑥 = 𝑦 as an antecedent. 𝜑 normally has 𝑧 free and can be read 𝜑(𝑧), and 𝜓 substitutes 𝑦 for 𝑧 and can be read 𝜑(𝑦). We do not require that 𝑥 and 𝑦 be distinct: if they are not, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with 𝑥𝑧, conjoin them, and apply dvelimdf 2333.

Other variants of this theorem are dvelimh 2334 (with no distinct variable restrictions) and dvelimhw 2171 (that avoids ax-13 2244). (Contributed by NM, 23-Nov-1994.)

(𝜑 → ∀𝑥𝜑)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelimv 2336* Similar to dvelim 2335 with first hypothesis replaced by a distinct variable condition. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
(𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelimnf 2337* Version of dvelim 2335 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
𝑥𝜑    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Theoremdveeq2ALT 2338* Alternate proof of dveeq2 2296, shorter but requiring ax-11 2032. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremax12OLD 2339 Obsolete proof of ax12 2302 as of 4-Jul-2021 . Rederivation of axiom ax-12 2045 from ax12v 2046, axc11r 2185, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12v2OLD 2340* Obsolete proof of ax12v 2046 as of 24-Mar-2021. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12a2OLD 2341* Obsolete proof of ax12v 2046 as of 24-Mar-2021. (Contributed by NM, 12-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremaxc15OLD 2342 Obsolete proof of axc15 2301 as of 24-Mar-2021. (Contributed by NM, 3-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12b 2343 A bidirectional version of axc15 2301. (Contributed by NM, 30-Jun-2006.)
((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremequvini 2344 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦. See equvinv 1957 for a shorter proof requiring fewer axioms when 𝑧 is required to be distinct from 𝑥 and 𝑦. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

Theoremequvel 2345 A variable elimination law for equality with no distinct variable requirements. Compare equvini 2344. (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Jun-2019.)
(∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)

Theoremequs5a 2346 A property related to substitution that unlike equs5 2349 does not require a distinctor antecedent. See equs5aALT 2175 for an alternate proof using ax-12 2045 but not ax13 2247. (Contributed by NM, 2-Feb-2007.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5e 2347 A property related to substitution that unlike equs5 2349 does not require a distinctor antecedent. See equs5eALT 2176 for an alternate proof using ax-12 2045 but not ax13 2247. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremequs45f 2348 Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2288 and does not require the non-freeness hypothesis. Theorem sb56 2148 replaces the non-freeness hypothesis with a dv condition and equs5 2349 replaces it with a distinctor as antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑦𝜑       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5 2349 Lemma used in proofs of substitution properties. If there is a dv condition on 𝑥, 𝑦, then sb56 2148 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2348 can be used. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb2 2350 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2427) or a non-freeness hypothesis (sb6f 2383). (Contributed by NM, 13-May-1993.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Theoremstdpc4 2351 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 𝑦 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "𝑥𝜑(𝑥) → 𝜑(𝑦), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥)." Axiom 4 of [Mendelson] p. 69. See also spsbc 3442 and rspsbc 3511. (Contributed by NM, 14-May-1993.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Theorem2stdpc4 2352 A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2351 for the analogous single specialization. See 2sp 2054 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)
(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremsb3 2353 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))

Theoremsb4 2354 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4a 2355 A version of sb4 2354 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb4b 2356 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremhbsb2 2357 Bound-variable hypothesis builder for substitution. (Contributed by NM, 14-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Theoremnfsb2 2358 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theoremhbsb2a 2359 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremsb4e 2360 One direction of a simplified definition of substitution that unlike sb4 2354 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremhbsb2e 2361 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Theoremhbsb3 2362 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 2363 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 2364* Alternate proof of axc16 2133, shorter but requiring ax-10 2017, ax-11 2032, ax-13 2244 and using df-nf 1708 and df-sb 1879. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremaxc16gALT 2365* Alternate proof of axc16g 2132 that uses df-sb 1879 and requires ax-10 2017, ax-11 2032, ax-13 2244. (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 2366 Substitution applied to an atomic wff. (Contributed by NM, 10-May-1993.)
[𝑦 / 𝑥]𝑥 = 𝑦

Theoremequsb2 2367 Substitution applied to an atomic wff. (Contributed by NM, 10-May-1993.)
[𝑦 / 𝑥]𝑦 = 𝑥

Theoremdveel1 2368* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))

Theoremdveel2 2369* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))

Theoremaxc14 2370 Axiom ax-c14 33995 is redundant if we assume ax-5 1837. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that 𝑤 is a dummy variable introduced in the proof. Its purpose is to satisfy the distinct variable requirements of dveel2 2369 and ax-5 1837. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))

Theoremdfsb2 2371 An alternate definition of proper substitution that, like df-sb 1879, 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 2372 An alternate definition of proper substitution df-sb 1879 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)))

Theoremsbequi 2373 An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Theoremsbequ 2374 An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Theoremdrsb1 2375 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))

Theoremdrsb2 2376 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, 27-Feb-2005.)
(∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Theoremsbft 2377 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
(Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))

Theoremsbf 2378 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥]𝜑𝜑)

Theoremsbh 2379 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥]𝜑𝜑)

Theoremsbf2 2380 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Theoremnfs1f 2381 If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥[𝑦 / 𝑥]𝜑

Theoremsb6x 2382 Equivalence involving substitution for a variable not free. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb6f 2383 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2350 and does not require the non-freeness hypothesis. Theorem sb6 2427 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑦𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5f 2384 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 1881 and does not require the non-freeness hypothesis. Theorem sb5 2428 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑦𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsbequ5 2385 Substitution does not change an identical variable specifier. (Contributed by NM, 15-May-1993.)
([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)

Theoremsbequ6 2386 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
([𝑤 / 𝑧] ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremnfsb4t 2387 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2388). (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 2388 A variable not free remains so after substitution 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)

Theoremsbn 2389 Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Theoremsbi1 2390 Removal of implication from substitution. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbi2 2391 Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)
(([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))

Theoremspsbim 2392 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbim 2393 Implication inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsbrim 2394 Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Theoremsblim 2395 Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜓       ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓))

Theoremsbor 2396 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))

Theoremsban 2397 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))

Theoremsb3an 2398 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
([𝑦 / 𝑥](𝜑𝜓𝜒) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓 ∧ [𝑦 / 𝑥]𝜒))

Theoremsbbi 2399 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

Theoremspsbbi 2400 Specialization of biconditional. (Contributed by NM, 2-Jun-1993.)
(∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))

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