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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremceqsalt 3201* Closed theorem version of ceqsalg 3203. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremceqsralt 3202* Restricted quantifier version of ceqsalt 3201. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremceqsalg 3203* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT 3204. (Contributed by NM, 29-Oct-2003.) (Proof shortened by BJ, 29-Sep-2019.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

TheoremceqsalgALT 3204* Alternate proof of ceqsalg 3203, not using ceqsalt 3201. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremceqsal 3205* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theoremceqsalv 3206* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theoremceqsralv 3207* Restricted quantifier version of ceqsalv 3206. (Contributed by NM, 21-Jun-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))

Theoremgencl 3208* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝐵))    &   (𝐴 = 𝐵 → (𝜑𝜓))    &   (𝜒𝜑)       (𝜃𝜓)

Theorem2gencl 3209* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐶𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐶)    &   (𝐷𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐷)    &   (𝐴 = 𝐶 → (𝜑𝜓))    &   (𝐵 = 𝐷 → (𝜓𝜒))    &   ((𝑥𝑅𝑦𝑅) → 𝜑)       ((𝐶𝑆𝐷𝑆) → 𝜒)

Theorem3gencl 3210* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
(𝐷𝑆 ↔ ∃𝑥𝑅 𝐴 = 𝐷)    &   (𝐹𝑆 ↔ ∃𝑦𝑅 𝐵 = 𝐹)    &   (𝐺𝑆 ↔ ∃𝑧𝑅 𝐶 = 𝐺)    &   (𝐴 = 𝐷 → (𝜑𝜓))    &   (𝐵 = 𝐹 → (𝜓𝜒))    &   (𝐶 = 𝐺 → (𝜒𝜃))    &   ((𝑥𝑅𝑦𝑅𝑧𝑅) → 𝜑)       ((𝐷𝑆𝐹𝑆𝐺𝑆) → 𝜃)

Theoremcgsexg 3211* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
(𝑥 = 𝐴𝜒)    &   (𝜒 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝜒𝜑) ↔ 𝜓))

Theoremcgsex2g 3212* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝜒)    &   (𝜒 → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(𝜒𝜑) ↔ 𝜓))

Theoremcgsex4g 3213* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → 𝜒)    &   (𝜒 → (𝜑𝜓))       (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤(𝜒𝜑) ↔ 𝜓))

Theoremceqsex 3214* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theoremceqsexv 3215* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)

Theoremceqsexv2d 3216* Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremceqsex2 3217* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
𝑥𝜓    &   𝑦𝜒    &   𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)

Theoremceqsex2v 3218* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵𝜑) ↔ 𝜒)

Theoremceqsex3v 3219* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))       (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)

Theoremceqsex4v 3220* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   (𝑤 = 𝐷 → (𝜃𝜏))       (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏)

Theoremceqsex6v 3221* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   (𝑤 = 𝐷 → (𝜃𝜏))    &   (𝑣 = 𝐸 → (𝜏𝜂))    &   (𝑢 = 𝐹 → (𝜂𝜁))       (∃𝑥𝑦𝑧𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁)

Theoremceqsex8v 3222* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V    &   𝐺 ∈ V    &   𝐻 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   (𝑤 = 𝐷 → (𝜃𝜏))    &   (𝑣 = 𝐸 → (𝜏𝜂))    &   (𝑢 = 𝐹 → (𝜂𝜁))    &   (𝑡 = 𝐺 → (𝜁𝜎))    &   (𝑠 = 𝐻 → (𝜎𝜌))       (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)

Theoremgencbvex 3223* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))       (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))

Theoremgencbvex2 3224* Restatement of gencbvex 3223 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 → ∃𝑥(𝜒𝐴 = 𝑦))       (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))

Theoremgencbval 3225* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)
𝐴 ∈ V    &   (𝐴 = 𝑦 → (𝜑𝜓))    &   (𝐴 = 𝑦 → (𝜒𝜃))    &   (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))       (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))

Theoremsbhypf 3226* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3518. (Contributed by Raph Levien, 10-Apr-2004.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))

Theoremvtoclgft 3227 Closed theorem form of vtoclgf 3237. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.)
(((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)

TheoremvtoclgftOLD 3228 Obsolete proof of vtoclgft 3227 as of 11-Aug-2021. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)

Theoremvtocldf 3229 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜓)    &   𝑥𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑𝜒)

Theoremvtocld 3230* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)

Theoremvtoclf 3231* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2250. (Contributed by NM, 30-Aug-1993.)
𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theoremvtocl 3232* Implicit substitution of a class for a setvar variable. See also vtoclALT 3233. (Contributed by NM, 30-Aug-1993.) Removed dependency on ax-10 2006. (Revised by BJ, 29-Nov-2020.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

TheoremvtoclALT 3233* Alternate proof of vtocl 3232. Shorter but requires more axioms. (Contributed by NM, 30-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       𝜓

Theoremvtocl2 3234* Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝜑       𝜓

Theoremvtocl3 3235* Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))    &   𝜑       𝜓

Theoremvtoclb 3236* Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝜑𝜓)       (𝜒𝜃)

Theoremvtoclgf 3237 Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theoremvtoclg1f 3238* Version of vtoclgf 3237 with one non-freeness hypothesis replaced with a dv condition, thus avoiding dependency on ax-11 2021 and ax-13 2234. (Contributed by BJ, 1-May-2019.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theoremvtoclg 3239* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝜑       (𝐴𝑉𝜓)

Theoremvtoclbg 3240* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝜑𝜓)       (𝐴𝑉 → (𝜒𝜃))

Theoremvtocl2gf 3241 Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝜓    &   𝑦𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝜑       ((𝐴𝑉𝐵𝑊) → 𝜒)

Theoremvtocl3gf 3242 Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴    &   𝑦𝐴    &   𝑧𝐴    &   𝑦𝐵    &   𝑧𝐵    &   𝑧𝐶    &   𝑥𝜓    &   𝑦𝜒    &   𝑧𝜃    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   𝜑       ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝜃)

Theoremvtocl2g 3243* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝜑       ((𝐴𝑉𝐵𝑊) → 𝜒)

Theoremvtoclgaf 3244* Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥𝐵𝜑)       (𝐴𝐵𝜓)

Theoremvtoclga 3245* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥𝐵𝜑)       (𝐴𝐵𝜓)

Theoremvtocl2gaf 3246* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝜓    &   𝑦𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   ((𝑥𝐶𝑦𝐷) → 𝜑)       ((𝐴𝐶𝐵𝐷) → 𝜒)

Theoremvtocl2ga 3247* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   ((𝑥𝐶𝑦𝐷) → 𝜑)       ((𝐴𝐶𝐵𝐷) → 𝜒)

Theoremvtocl3gaf 3248* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝐴    &   𝑦𝐴    &   𝑧𝐴    &   𝑦𝐵    &   𝑧𝐵    &   𝑧𝐶    &   𝑥𝜓    &   𝑦𝜒    &   𝑧𝜃    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝑥𝑅𝑦𝑆𝑧𝑇) → 𝜑)       ((𝐴𝑅𝐵𝑆𝐶𝑇) → 𝜃)

Theoremvtocl3ga 3249* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜃))    &   ((𝑥𝐷𝑦𝑅𝑧𝑆) → 𝜑)       ((𝐴𝐷𝐵𝑅𝐶𝑆) → 𝜃)

Theoremvtocl4g 3250* Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜌))    &   (𝑤 = 𝐷 → (𝜌𝜃))    &   𝜑       (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃)

Theoremvtocl4ga 3251* Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑧 = 𝐶 → (𝜒𝜌))    &   (𝑤 = 𝐷 → (𝜌𝜃))    &   (((𝑥𝑄𝑦𝑅) ∧ (𝑧𝑆𝑤𝑇)) → 𝜑)       (((𝐴𝑄𝐵𝑅) ∧ (𝐶𝑆𝐷𝑇)) → 𝜃)

Theoremvtocleg 3252* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.)
(𝑥 = 𝐴𝜑)       (𝐴𝑉𝜑)

Theoremvtoclegft 3253* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3254.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)

Theoremvtoclef 3254* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑    &   𝐴 ∈ V    &   (𝑥 = 𝐴𝜑)       𝜑

Theoremvtocle 3255* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝜑)       𝜑

Theoremvtoclri 3256* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝑥𝐵 𝜑       (𝐴𝐵𝜓)

Theoremspcimgft 3257 A closed version of spcimgf 3259. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))

Theoremspcgft 3258 A closed version of spcgf 3261. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
𝑥𝜓    &   𝑥𝐴       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))

Theoremspcimgf 3259 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))

Theoremspcimegf 3260 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜓𝜑))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))

Theoremspcgf 3261 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))

Theoremspcegf 3262 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
𝑥𝐴    &   𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))

Theoremspcimdv 3263* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

Theoremspcdv 3264* Rule of specialization, using implicit substitution. Analogous to rspcdv 3285. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓𝜒))

Theoremspcimedv 3265* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝜓))

Theoremspcgv 3266* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝜑𝜓))

Theoremspcegv 3267* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))

Theoremspc2egv 3268* Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝜓 → ∃𝑥𝑦𝜑))

Theoremspc2gv 3269* Specialization with two quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))

Theoremspc3egv 3270* Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))

Theoremspc3gv 3271* Specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))

Theoremspcv 3272* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspcev 3273* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝜓 → ∃𝑥𝜑)

Theoremspc2ev 3274* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (𝜓 → ∃𝑥𝑦𝜑)

Theoremrspct 3275* A closed version of rspc 3276. (Contributed by Andrew Salmon, 6-Jun-2011.)
𝑥𝜓       (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))

Theoremrspc 3276* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))

Theoremrspce 3277* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Theoremrspcv 3278* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))

Theoremrspccv 3279* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))

Theoremrspcva 3280* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → 𝜓)

Theoremrspccva 3281* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((∀𝑥𝐵 𝜑𝐴𝐵) → 𝜓)

Theoremrspcev 3282* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Theoremrspcimdv 3283* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))

Theoremrspcimedv 3284* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))

Theoremrspcdv 3285* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓𝜒))

Theoremrspcedv 3286* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))

Theoremrspcda 3287* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)
(𝑥 = 𝐶 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝐶𝐴)    &   𝑥𝜑       (𝜑𝜒)

Theoremrspcdva 3288* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝑥 = 𝐶 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝐴 𝜓)    &   (𝜑𝐶𝐴)       (𝜑𝜒)

Theoremrspcedvd 3289* Restricted existential specialization, using implicit substitution. Variant of rspcedv 3286. (Contributed by AV, 27-Nov-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝜒)       (𝜑 → ∃𝑥𝐵 𝜓)

Theoremrspcedeq1vd 3290* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3289 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)

Theoremrspcedeq2vd 3291* Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3289 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 = 𝐴) → 𝐶 = 𝐷)       (𝜑 → ∃𝑥𝐵 𝐶 = 𝐷)

Theoremrspc2 3292* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
𝑥𝜒    &   𝑦𝜓    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))

Theoremrspc2v 3293* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))

Theoremrspc2va 3294* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜑) → 𝜓)

Theoremrspc2ev 3295* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜓))       ((𝐴𝐶𝐵𝐷𝜓) → ∃𝑥𝐶𝑦𝐷 𝜑)

Theoremrspc3v 3296* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜃))    &   (𝑧 = 𝐶 → (𝜃𝜓))       ((𝐴𝑅𝐵𝑆𝐶𝑇) → (∀𝑥𝑅𝑦𝑆𝑧𝑇 𝜑𝜓))

Theoremrspc3ev 3297* 3-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)
(𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜒𝜃))    &   (𝑧 = 𝐶 → (𝜃𝜓))       (((𝐴𝑅𝐵𝑆𝐶𝑇) ∧ 𝜓) → ∃𝑥𝑅𝑦𝑆𝑧𝑇 𝜑)

Theoremralxpxfr2d 3298* Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
𝐴 ∈ V    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶𝑧𝐷 𝜒))

Theoremrexraleqim 3299* Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)
(𝑥 = 𝑧 → (𝜓𝜑))    &   (𝑧 = 𝑌 → (𝜑𝜃))       ((∃𝑧𝐴 𝜑 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → 𝜃)

Theoremeqvinc 3300* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝐴 ∈ V       (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))

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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 42301-42360
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