Home Metamath Proof ExplorerTheorem List (p. 22 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >