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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.41 2101 Theorem 19.41 of [Margaris] p. 90. See 19.41v 1912 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))
 
Theorem19.42-1 2102 One direction of 19.42 2103. (Contributed by Wolf Lammen, 10-Jul-2021.)
𝑥𝜑       ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.42 2103 Theorem 19.42 of [Margaris] p. 90. See 19.42v 1916 for a version requiring fewer axioms. See exan 1786 for an immediate version. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
 
Theorem19.44 2104 Theorem 19.44 of [Margaris] p. 90. See 19.44v 1910 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.45 2105 Theorem 19.45 of [Margaris] p. 90. See 19.45v 1911 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
 
Theoremequsalv 2106* Version of equsal 2289 with a dv condition, which does not require ax-13 2244. See equsalvw 1929 for a version with two dv conditions requiring fewer axioms. See also the dual form equsexv 2107. (Contributed by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremequsexv 2107* Version of equsex 2290 with a dv condition, which does not require ax-13 2244. See equsexvw 1930 for a version with two dv conditions requiring fewer axioms. See also the dual form equsalv 2106. (Contributed by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremsbequ1 2108 An equality theorem for substitution. (Contributed by NM, 16-May-1993.)
(𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
 
Theoremsbequ12 2109 An equality theorem for substitution. (Contributed by NM, 14-May-1993.)
(𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
 
Theoremsbequ12r 2110 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
 
Theoremsbequ12a 2111 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 2112 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)
 
Theoremspimv1 2113* Version of spim 2252 with a dv condition, which does not require ax-13 2244. See spimvw 1925 for a version with two dv conditions, requiring fewer axioms, and spimv 2255 for another variant. (Contributed by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremnf5 2114 Alternate definition of df-nf 1708. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1708 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
 
Theoremnf6 2115 An alternate definition of df-nf 1708. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))
 
Theoremnf5d 2116 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 2117 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 2118 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 2119 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See also 19.21 2073 and 19.21v 1866. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜑 → ∀𝑥𝜑)       (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
 
Theorem19.23h 2120 Theorem 19.23 of [Margaris] p. 90. See 19.23 2078. (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))
 
Theoremequsalhw 2121* Weaker version of equsalh 2292 with a dv condition which does not require ax-13 2244. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremequsexhv 2122* Version of equsexh 2293 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 31-May-2019.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
Theoremhbim1 2123 A closed form of hbim 2125. (Contributed by NM, 2-Jun-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theoremhbimd 2124 Deduction form of bound-variable hypothesis builder hbim 2125. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
 
Theoremhbim 2125 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theoremhban 2126 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)       ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 
Theoremhb3an 2127 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)       ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
 
Theoremaxc4 2128 Show that the original axiom ax-c4 33988 can be derived from ax-4 1735 (alim 1736), ax-10 2017 (hbn1 2018), sp 2051 and propositional calculus. See ax4fromc4 33998 for the rederivation of ax-4 1735 from ax-c4 33988.

Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.)

(∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theoremaxc4i 2129 Inference version of axc4 2128. (Contributed by NM, 3-Jan-1993.)
(∀𝑥𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)
 
Theoremaxc7 2130 Show that the original axiom ax-c7 33989 can be derived from ax-10 2017 (hbn1 2018) , sp 2051 and propositional calculus. See ax10fromc7 33999 for the rederivation of ax-10 2017 from ax-c7 33989.

Normally, axc7 2130 should be used rather than ax-c7 33989, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.)

(¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
 
Theoremaxc7e 2131 Abbreviated version of axc7 2130 using the existential quantifier. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑥𝜑𝜑)
 
Theoremaxc16g 2132* Generalization of axc16 2133. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 1933 }, theorem ax12v 2046. (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 2244, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2137. (Revised by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 
Theoremaxc16 2133* Proof of older axiom ax-c16 33996. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 
Theoremaxc16gb 2134* Biconditional strengthening of axc16g 2132. (Contributed by NM, 15-May-1993.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
 
Theoremaxc16nf 2135* If dtru 4848 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 2032. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2017. (Revised by Wolf lammen, 12-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 
Theoremaxc11v 2136* Version of axc11 2312 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 1933 }, from ax12v 2046 (contrary to axc11 2312 which seems to require the full ax-12 2045 and ax-13 2244). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
Theoremaxc11rv 2137* Version of axc11r 2185 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 1933 }, from ax12v 2046 (contrary to axc11 2312 which seems to require the full ax-12 2045). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
 
Theoremaxc11rvOLD 2138* Obsolete proof of axc11rv 2137 as of 11-Oct-2021. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
 
Theoremaxc11vOLD 2139* Obsolete proof of axc11v 2136 as of 11-Oct-2021. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
Theoremmodal-b 2140 The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.)
(𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
 
Theorem19.9ht 2141 A closed version of 19.9 2070. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
 
Theoremhbnt 2142 Closed theorem version of bound-variable hypothesis builder hbn 2144. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof shortened by Wolf Lammen, 14-Oct-2021.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
 
TheoremhbntOLD 2143 Obsolete proof of hbnt 2142 as of 13-Oct-2021. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
 
Theoremhbn 2144 If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)
(𝜑 → ∀𝑥𝜑)       𝜑 → ∀𝑥 ¬ 𝜑)
 
Theoremhbnd 2145 Deduction form of bound-variable hypothesis builder hbn 2144. (Contributed by NM, 3-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
 
Theoremexlimih 2146 Inference associated with 19.23 2078. See exlimiv 1856 for a version with a dv 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 2147 Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.)
(𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))
 
Theoremsb56 2148* Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1879. The implication "to the left" is equs4 2288 and does not require any dv condition (but the version with a dv condition, equs4v 1928, requires fewer axioms). Theorem equs45f 2348 replaces the dv condition with a non-freeness hypothesis and equs5 2349 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2107 in place of equsex 2290 in order to remove dependency on ax-13 2244. (Revised by BJ, 20-Dec-2020.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremhba1 2149 The setvar 𝑥 is not free in 𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 12-Oct-2021.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 
TheoremhbexOLD 2150 Obsolete proof of hbex 2154 as of 16-Oct-2021. (Contributed by NM, 12-Mar-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
 
Theoremnfal 2151 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥𝑦𝜑
 
Theoremnfex 2152 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Reduce symbol count in nfex 2152, hbex 2154. (Revised by Wolf Lammen, 16-Oct-2021.)
𝑥𝜑       𝑥𝑦𝜑
 
TheoremnfexOLD 2153 Obsolete proof of nfex 2152 as of 16-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       𝑥𝑦𝜑
 
Theoremhbex 2154 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2152, hbex 2154. (Revised by Wolf Lammen, 16-Oct-2021.)
(𝜑 → ∀𝑥𝜑)       (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
 
Theoremnfa1OLD 2155 Obsolete proof of nfa1 2026 as of 12-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1708 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑
 
Theoremnfnf 2156 If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
𝑥𝜑       𝑥𝑦𝜑
 
Theoremnfnf1OLD 2157 Obsolete proof of nfnf1 2029 as of 12-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑥𝜑
 
Theoremaxc11nlemOLD 2158* Obsolete proof of axc11nlemOLD2 1986 as of 14-Mar-2021. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))       (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)
 
Theoremaxc16gOLD 2159* Obsolete proof of axc16g 2132 as of 11-Oct-2021. (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 2244, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2137. (Revised by Wolf Lammen, 11-Oct-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 
TheoremaevOLD 2160* Obsolete proof of aev 1981 as of 19-Mar-2021. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2032. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2244, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
 
Theoremaxc16nfOLD 2161* Obsolete proof of axc16nf 2135 as of 12-Oct-2021. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2032. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 
Theorem19.12 2162 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 2178 and r19.12sn 4246. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
(∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
Theoremnfald 2163 Deduction form of nfal 2151. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) (Proof shortened by Wolf Lammen, 16-Oct-2021.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
TheoremnfaldOLD 2164 Obsolete proof of nfald 2163 as of 16-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
Theoremnfexd 2165 If 𝑥 is not free in 𝜓, it is not free in 𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
Theoremnfa2OLD 2166 Obsolete proof of nfa2 2038 as of 18-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝑦𝑥𝜑
 
TheoremexanOLDOLD 2167 Obsolete proof of exan 1786 as of 7-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)
 
Theoremaaan 2168 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
 
Theoremeeor 2169 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
 
Theoremcbv3v 2170* Version of cbv3 2263 with a dv condition, which does not require ax-13 2244. (Contributed by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)
 
Theoremdvelimhw 2171* Proof of dvelimh 2334 without using ax-13 2244 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
 
Theoremcbv3hv 2172* Version of cbv3h 2264 with a dv condition on 𝑥, 𝑦, which does not require ax-13 2244. Was used in a proof of axc11n 2305 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.)
(𝜑 → ∀𝑦𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)
 
Theoremcbvalv1 2173* Version of cbval 2269 with a dv condition, which does not require ax-13 2244. See cbvalvw 1967 for a version with two dv conditions, requiring fewer axioms, and cbvalv 2271 for another variant. (Contributed by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
Theoremcbvexv1 2174* Version of cbvex 2270 with a dv condition, which does not require ax-13 2244. See cbvexvw 1968 for a version with two dv conditions, requiring fewer axioms, and cbvexv 2273 for another variant. (Contributed by BJ, 31-May-2019.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
Theoremequs5aALT 2175 Alternate proof of equs5a 2346. Uses ax-12 2045 but not ax-13 2244. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
 
Theoremequs5eALT 2176 Alternate proof of equs5e 2347. Uses ax-12 2045 but not ax-13 2244. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
 
Theorempm11.53 2177* Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1904 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
 
Theorem19.12vv 2178* Special case of 19.12 2162 where its converse holds. See 19.12vvv 1905 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))
 
Theoremeean 2179 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theoremeeanv 2180* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 
Theoremeeeanv 2181* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
 
Theoremee4anv 2182* Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
(∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
 
TheoremcleljustALT 2183* Alternate proof of cleljust 1996. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how DV conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 
TheoremcleljustALT2 2184* Alternate proof of cleljust 1996. Compared with cleljustALT 2183, it uses nfv 1841 followed by equsexv 2107 instead of ax-5 1837 followed by equsexhv 2122, so it uses the idiom 𝑥𝜑 instead of 𝜑 → ∀𝑥𝜑 to express non-freeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 
Theoremaxc11r 2185 Same as axc11 2312 but with reversed antecedent. Note the use of ax-12 2045 (and not merely ax12v 2046). (Contributed by NM, 25-Jul-2015.)
(∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
TheoremnfrOLD 2186 Obsolete proof of nf5r 2062 as of 6-Oct-2021. (Contributed by Mario Carneiro, 26-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
TheoremnfriOLD 2187 Obsolete proof of nf5ri 2063 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (𝜑 → ∀𝑥𝜑)
 
TheoremnfrdOLD 2188 Obsolete proof of nf5rd 2064 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → (𝜓 → ∀𝑥𝜓))
 
TheoremalimdOLD 2189 Obsolete proof of alimd 2079 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
TheoremalrimiOLD 2190 Obsolete proof of alrimi 2080 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)
 
TheoremnfdOLD 2191 Obsolete proof of nf5d 2116 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
TheoremnfdhOLD 2192 Obsolete proof of nf5dh 2024 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
TheoremalrimddOLD 2193 Obsolete proof of alrimdd 2081 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
TheoremalrimdOLD 2194 Obsolete proof of alrimd 2082 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
TheoremeximdOLD 2195 Obsolete proof of eximd 2083 as of 6-Oct-2021. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
TheoremnexdOLD 2196 Obsolete proof of nexd 2087 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
TheoremalbidOLD 2197 Obsolete proof of albid 2088 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
TheoremexbidOLD 2198 Obsolete proof of exbid 2089 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
TheoremnfbidfOLD 2199 Obsolete proof of nfbidf 2090 as of 6-Oct-2021. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
 
Theorem19.3OLD 2200 Obsolete proof of 19.3 2067 as of 6-Oct-2021. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑       (∀𝑥𝜑𝜑)
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