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Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxc11nOLDOLD 2201 Old proof of axc11n 2199. 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 1936. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 
Theoremaxc11nALT 2202 Alternate proof of axc11n 2199 from axc11nlemALT 2198. (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 2203 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 2204 A commutation rule for identical variable specifiers. (Contributed by NM, 10-May-1993.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)
 
Theoremnaecoms 2205 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)
 
Theoremaxc11 2206 Show that ax-c11 33080 can be derived from ax-c11n 33081 in the form of axc11n 2199. Normally, axc11 2206 should be used rather than ax-c11 33080, 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 2207 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 2208 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧𝑥 𝑥 = 𝑦
 
Theoremhbnae 2209 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 2210 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦
 
Theoremhbnaes 2211 Rule that applies hbnae 2209 to antecedent. (Contributed by NM, 15-May-1993.)
(∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑥 𝑥 = 𝑦𝜑)
 
TheoremaevlemALTOLD 2212* Older alternate version of aevlem 1929. 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 2213* Older alternate proof of aev 1931. 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 2214* Inference with axc16 2072 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
Theoremaxc16nfALT 2215* Alternate proof of axc16nf 2075, shorter but requiring ax-11 1971. (Contributed by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 
Theoremdral2 2216 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 2217. (Revised by Wolf Lammen, 4-Mar-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
 
Theoremdral1 2217 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 1971. (Revised by Wolf Lammen, 6-Sep-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
Theoremdral1ALT 2218 Alternate proof of dral1 2217, shorter but requiring ax-11 1971. (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 2219 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 2220 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 2221 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 2222 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 2223 Variation on nfald 2083 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 2224 Variation on nfexd 2084 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 2225 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 2226 Version of dvelimv 2230 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 2227 Deduction form of dvelimf 2226. (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 2228 Version of dvelim 2229 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 2229* 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 2227.

Other variants of this theorem are dvelimh 2228 (with no distinct variable restrictions) and dvelimhw 2111 (that avoids ax-13 2137). (Contributed by NM, 23-Nov-1994.)

(𝜑 → ∀𝑥𝜑)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremdvelimv 2230* Similar to dvelim 2229 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 2231* Version of dvelim 2229 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
𝑥𝜑    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
 
Theoremdveeq2ALT 2232* Alternate proof of dveeq2 2190, shorter but requiring ax-11 1971. (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 2233 Obsolete proof of ax12 2196 as of 4-Jul-2021 . Rederivation of axiom ax-12 1983 from ax12v 1984, axc11r 2136, 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 2234* Obsolete proof of ax12v 1984 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 2235* Obsolete proof of ax12v 1984 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 2236 Obsolete proof of axc15 2195 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 2237 A bidirectional version of axc15 2195. (Contributed by NM, 30-Jun-2006.)
((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremequvini 2238 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 1909 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 2239 A variable elimination law for equality with no distinct variable requirements. Compare equvini 2238. (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 2240 A property related to substitution that unlike equs5 2243 does not require a distinctor antecedent. See equs5aALT 2119 for an alternate proof using ax-12 1983 but not ax13 2140. (Contributed by NM, 2-Feb-2007.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremequs5e 2241 A property related to substitution that unlike equs5 2243 does not require a distinctor antecedent. See equs5eALT 2120 for an alternate proof using ax-12 1983 but not ax13 2140. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 
Theoremequs45f 2242 Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2181 and does not require the non-freeness hypothesis. Theorem sb56 2128 replaces the non-freeness hypothesis with a dv condition and equs5 2243 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 2243 Lemma used in proofs of substitution properties. If there is a dv condition on 𝑥, 𝑦, then sb56 2128 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2242 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 2244 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2321) or a non-freeness hypothesis (sb6f 2277). (Contributed by NM, 13-May-1993.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
 
Theoremstdpc4 2245 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 3319 and rspsbc 3388. (Contributed by NM, 14-May-1993.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
 
Theorem2stdpc4 2246 A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2245 for the analogous single specialization. See 2sp 1997 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)
(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
 
Theoremsb3 2247 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 2248 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 2249 A version of sb4 2248 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremsb4b 2250 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
 
Theoremhbsb2 2251 Bound-variable hypothesis builder for substitution. (Contributed by NM, 14-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
 
Theoremnfsb2 2252 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
 
Theoremhbsb2a 2253 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
 
Theoremsb4e 2254 One direction of a simplified definition of substitution that unlike sb4 2248 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 
Theoremhbsb2e 2255 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
 
Theoremhbsb3 2256 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 2257 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 2258* Alternate proof of axc16 2072, shorter but requiring ax-11 1971 and using df-sb 1831. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
Theoremaxc16gALT 2259* Alternate proof of axc16g 2071 that uses df-sb 1831. (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 2260 Substitution applied to an atomic wff. (Contributed by NM, 10-May-1993.)
[𝑦 / 𝑥]𝑥 = 𝑦
 
Theoremequsb2 2261 Substitution applied to an atomic wff. (Contributed by NM, 10-May-1993.)
[𝑦 / 𝑥]𝑦 = 𝑥
 
Theoremdveel1 2262* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))
 
Theoremdveel2 2263* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))
 
Theoremaxc14 2264 Axiom ax-c14 33084 is redundant if we assume ax-5 1793. 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 2263 and ax-5 1793. 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 2265 An alternate definition of proper substitution that, like df-sb 1831, 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 2266 An alternate definition of proper substitution df-sb 1831 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 2267 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 2268 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 2269 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 2270 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 2271 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 2272 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 2273 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 2274 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)
 
Theoremnfs1f 2275 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 2276 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 2277 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the left" is sb2 2244 and does not require the non-freeness hypothesis. Theorem sb6 2321 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 2278 Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 1833 and does not require the non-freeness hypothesis. Theorem sb5 2322 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 2279 Substitution does not change an identical variable specifier. (Contributed by NM, 15-May-1993.)
([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
 
Theoremsbequ6 2280 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)
([𝑤 / 𝑧] ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
 
Theoremnfsb4t 2281 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2282). (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 2282 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 2283 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 2284 Removal of implication from substitution. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 
Theoremsbi2 2285 Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)
(([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑𝜓))
 
Theoremspsbim 2286 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 2287 Implication inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.)
([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 
Theoremsbrim 2288 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 2289 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 2290 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 2291 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 2292 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 2293 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 2294 Specialization of biconditional. (Contributed by NM, 2-Jun-1993.)
(∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
 
Theoremsbbid 2295 Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
 
Theoremsblbis 2296 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜒𝜑) ↔ ([𝑦 / 𝑥]𝜒𝜓))
 
Theoremsbrbis 2297 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒))
 
Theoremsbrbif 2298 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜒    &   ([𝑦 / 𝑥]𝜑𝜓)       ([𝑦 / 𝑥](𝜑𝜒) ↔ (𝜓𝜒))
 
Theoremsbequ8ALT 2299 Alternate proof of sbequ8 1835, shorter but requiring more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
 
Theoremsbie 2300 Conversion of implicit substitution to explicit substitution. (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)
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