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Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfim1 2201 A closed form of nfim 1898. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theoremnfan1 2202 A closed form of nfan 1901. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 7-Jul-2022.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       𝑥(𝜑𝜓)

Theorem19.3t 2203 Closed form of 19.3 2204 and version of 19.9t 2206 with a universal quantifier. (Contributed by NM, 9-Nov-2020.) (Proof shortened by BJ, 9-Oct-2022.)
(Ⅎ𝑥𝜑 → (∀𝑥𝜑𝜑))

Theorem19.3 2204 A wff may be quantified with a variable not free in it. Version of 19.9 2207 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1987 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥𝜑𝜑)

Theorem19.9d 2205 A deduction version of one direction of 19.9 2207. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))

Theorem19.9t 2206 Closed form of 19.9 2207 and version of 19.3t 2203 with an existential quantifier. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
(Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))

Theorem19.9 2207 A wff may be existentially quantified with a variable not free in it. Version of 19.3 2204 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1989 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
𝑥𝜑       (∃𝑥𝜑𝜑)

Theorem19.21t 2208 Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2209. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
(Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Theorem19.21 2209 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1941 for a version requiring fewer axioms. See also 19.21h 2297. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theoremstdpc5 2210 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis 𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑". With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 2021. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1940 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2146. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))

Theorem19.21-2 2211 Version of 19.21 2209 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))

Theorem19.23t 2212 Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2213. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 8-Oct-2022.)
(Ⅎ𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorem19.23 2213 Theorem 19.23 of [Margaris] p. 90. See 19.23v 1944 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremalimd 2214 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1812. See alimdh 1819, alimdv 1918 for variants requiring fewer axioms. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalrimi 2215 Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2209. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremalrimdd 2216 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2209. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremalrimd 2217 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2209. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremeximd 2218 Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1835. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremexlimi 2219 Inference associated with 19.23 2213. See exlimiv 1932 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜓    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremexlimd 2220 Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

Theoremexlimimdd 2221 Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2222. (Revised by Wolf Lammen, 3-Sep-2023.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremexlimdd 2222 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

TheoremexlimddOLD 2223 Obsolete version of exlimdd 2222 as of 3-Sep-2023. (Contributed by Mario Carneiro, 9-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)

TheoremexlimimddOLD 2224 Obsolete version of exlimimdd 2221 as of 3-Sep-2023. (Contributed by ML, 17-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)

Theoremnexd 2225 Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremalbid 2226 Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbid 2227 Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremnfbidf 2228 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Theorem19.16 2229 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Theorem19.17 2230 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorem19.27 2231 Theorem 19.27 of [Margaris] p. 90. See 19.27v 1997 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28 2232 Theorem 19.28 of [Margaris] p. 90. See 19.28v 1998 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.19 2233 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Theorem19.36 2234 Theorem 19.36 of [Margaris] p. 90. See 19.36v 1995 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.36i 2235 Inference associated with 19.36 2234. See 19.36iv 1948 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
𝑥𝜓    &   𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem19.37 2236 Theorem 19.37 of [Margaris] p. 90. See 19.37v 1999 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Theorem19.32 2237 Theorem 19.32 of [Margaris] p. 90. See 19.32v 1942 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑       (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Theorem19.31 2238 Theorem 19.31 of [Margaris] p. 90. See 19.31v 1943 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.41 2239 Theorem 19.41 of [Margaris] p. 90. See 19.41v 1951 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.42 2240 Theorem 19.42 of [Margaris] p. 90. See 19.42v 1955 for a version requiring fewer axioms. See exan 1863 for an immediate version. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theorem19.44 2241 Theorem 19.44 of [Margaris] p. 90. See 19.44v 2000 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45 2242 Theorem 19.45 of [Margaris] p. 90. See 19.45v 2001 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theoremspimfv 2243* Specialization, using implicit substitution. Version of spim 2407 with a disjoint variable condition, which does not require ax-13 2392. See spimvw 2003 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2410 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremchvarfv 2244* Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 2415 with a disjoint variable condition, which does not require ax-13 2392. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremcbv3v2 2245* Version of cbv3 2417 with two disjoint variable conditions, which does not require ax-11 2162 nor ax-13 2392. (Contributed by BJ, 24-Jun-2019.) (Proof shortened by Wolf Lammen, 30-Aug-2021.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremsb4av 2246* Version of sb4a 2511 with a disjoint variable condition, which does not require ax-13 2392. The distinctor antecedent from sb4b 2501 is replaced by a disjoint variable condition in this theorem. (Contributed by NM, 2-Feb-2007.) (Revised by BJ, 15-Dec-2023.)
([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))

Theoremsbimd 2247 Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2071. (Revised by Steven Nguyen, 9-Jul-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))

Theoremsbbid 2248 Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2146 and ax-13 2392. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2071. (Revised by Steven Nguyen, 11-Jul-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theorem2sbbid 2249 Deduction doubly substituting both sides of a biconditional. (Contributed by AV, 30-Jul-2023.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))    &   𝑦𝜑       (𝜑 → ([𝑡 / 𝑥][𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜒))

TheoremsbbidOLD 2250 Obsolete version of sbbid 2248 as of 10-Jul-2023. Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2146 and ax-13 2392. (Revised by Wolf Lammen, 24-Nov-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Theoremsbequ1 2251 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2071. (Revised by BJ, 22-Dec-2020.)
(𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))

Theoremsbequ2 2252 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2071. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.)
(𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))

Theoremsbequ2OLD 2253 Obsolete version of sbequ2 2252 as of 3-Feb-2024. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) Revise df-sb 2071. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))

Theoremstdpc7 2254 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 2036.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑥)". Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Theoremsbequ12 2255 An equality theorem for substitution. (Contributed by NM, 14-May-1993.)
(𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))

Theoremsbequ12r 2256 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Theoremsbelx 2257* Elimination of substitution. Also see sbel2x 2500. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2392. (Revised by Wolf Lammen, 6-Aug-2023.) Avoid ax-10 2146. (Revised by Gino Giotto, 20-Aug-2023.)
(𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))

Theoremsbequ12a 2258 An equality theorem for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Jun-2019.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

Theoremsbid 2259 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)
([𝑥 / 𝑥]𝜑𝜑)

Theoremsbcov 2260* A composition law for substitution. Version of sbco 2551 with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Gino Giotto, 7-Aug-2023.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Theoremsb6a 2261* Equivalence for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Sep-2018.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))

Theoremsbid2vw 2262* Reverting substitution yields the original expression. Based on fewer axioms than sbid2v 2553, at the expense of an extra distinct variable condition. (Contributed by NM, 14-May-1993.) (Revised by Wolf Lammen, 5-Aug-2023.)
([𝑡 / 𝑥][𝑥 / 𝑡]𝜑𝜑)

Theoremaxc16g 2263* Generalization of axc16 2264. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2016 }, theorem ax12v 2180. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2392, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2268. (Revised by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Theoremaxc16 2264* Proof of older axiom ax-c16 36133. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremaxc16gb 2265* Biconditional strengthening of axc16g 2263. (Contributed by NM, 15-May-1993.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))

Theoremaxc16nf 2266* If dtru 5258 is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2162. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2146. (Revised by Wolf lammen, 12-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theoremaxc11v 2267* Version of axc11 2454 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2016 }, from ax12v 2180 (contrary to axc11 2454 which seems to require the full ax-12 2179 and ax-13 2392). (Contributed by NM, 16-May-2008.) (Revised by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremaxc11rv 2268* Version of axc11r 2388 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2016 }, from ax12v 2180 (contrary to axc11 2454 which seems to require the full ax-12 2179 and ax-13 2392, and to axc11r 2388 which seems to require the full ax-12 2179). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))

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

Theoremequsalv 2270* An equivalence related to implicit substitution. Version of equsal 2441 with a disjoint variable condition, which does not require ax-13 2392. See equsalvw 2011 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2271. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexv 2271* An equivalence related to implicit substitution. Version of equsex 2442 with a disjoint variable condition, which does not require ax-13 2392. See equsexvw 2012 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2270. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremsbft 2272 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 2273 Substitution for a variable not free in a wff does not affect it. For a version requiring disjoint variables but fewer axioms, see sbv 2099. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑥𝜑       ([𝑦 / 𝑥]𝜑𝜑)

Theoremsbf2 2274 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)

Theoremsbh 2275 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.)
(𝜑 → ∀𝑥𝜑)       ([𝑦 / 𝑥]𝜑𝜑)

Theoremhbs1 2276* The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremnfs1f 2277 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

Theoremsb5 2278* Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2096 and even needs no disjoint variable condition, see sb1 2505. Theorem sb5f 2540 replaces the disjoint variable condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Shorten sb56 2279. (Revised by Wolf Lammen, 4-Sep-2023.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsb56 2279* Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct, namely, alternate definitions sb5 2278 and sb6 2094. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2071. The implication "to the left" is equs4 2440 and does not require any disjoint variable condition (but the version with a disjoint variable condition, equs4v 2007, requires fewer axioms). Theorem equs45f 2484 replaces the disjoint variable condition with a non-freeness hypothesis and equs5 2485 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2271 in place of equsex 2442 in order to remove dependency on ax-13 2392. (Revised by BJ, 20-Dec-2020.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb56OLD 2280* Obsolete version of sb56 2279 as of 4-Sep-2023. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2271 in place of equsex 2442 in order to remove dependency on ax-13 2392. (Revised by BJ, 20-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5av 2281* A property related to substitution that replaces the distinctor from equs5 2485 to a disjoint variable condition. Version of equs5a 2482 with a disjoint variable condition, which does not require ax-13 2392. See also sb56 2279. (Contributed by NM, 2-Feb-2007.) (Revised by Gino Giotto, 15-Dec-2023.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb6OLD 2282* Obsolete version of sb6 2094 as of 7-Jul-2023. Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left", sb2vOLD 2098, also holds without a disjoint variable condition (sb2 2506). Theorem sb6f 2539 replaces the disjoint variable condition with a non-freeness hypothesis. Theorem sb4b 2501 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb5OLD 2283* Obsolete version of sb5 2278 as of 4-Sep-2023.) (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theorem2sb5 2284* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))

Theoremsbco4lem 2285* Lemma for sbco4 2286. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.)
([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremsbco4 2286* Two ways of exchanging two variables. Both sides of the biconditional exchange 𝑥 and 𝑦, either via two temporary variables 𝑢 and 𝑣, or a single temporary 𝑤. (Contributed by Jim Kingdon, 25-Sep-2018.)
([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremdfsb7 2287* An alternate definition of proper substitution df-sb 2071. By introducing a dummy variable 𝑦 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑡, 𝑥, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑦 effectively insulates 𝑡 from 𝑥. To achieve this, we use a chain of two substitutions in the form of sb5 2278, first 𝑦 for 𝑥 then 𝑡 for 𝑦. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2803. Theorem sb7h 2571 provides a version where 𝜑 and 𝑦 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2071. (Revised by BJ, 25-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremdfsb7OLD 2288* Obsolete version of dfsb7 2287 as of 3-Sep-2023. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2071. (Revised by BJ, 25-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremsbn 2289 Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) Revise df-sb 2071. (Revised by BJ, 25-Dec-2020.)
([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)

Theoremsbex 2290* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)

TheoremsbbibOLD 2291* Obsolete version of sbbib 2382 as of 4-Sep-2023. (Contributed by AV, 6-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓       (∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Theoremnf5 2292 Alternate definition of df-nf 1786. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))

Theoremnf6 2293 An alternate definition of df-nf 1786. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremnf5d 2294 Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)

Theoremnf5di 2295 Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either 𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)
(𝜑 → Ⅎ𝑥𝜑)       𝑥𝜑

Theorem19.9h 2296 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥𝜑𝜑)

Theorem19.21h 2297 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See also 19.21 2209 and 19.21v 1941. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜑 → ∀𝑥𝜑)       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Theorem19.23h 2298 Theorem 19.23 of [Margaris] p. 90. See 19.23 2213. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜓 → ∀𝑥𝜓)       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theoremexlimih 2299 Inference associated with 19.23 2213. See exlimiv 1932 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜓 → ∀𝑥𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theoremexlimdh 2300 Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.)
(𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))

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