Theorem vtoclgf 3264

Description: 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.)

Hypotheses

Ref	Expression
vtoclgf.1	|- F/_ x A
vtoclgf.2	|- F/ x ps
vtoclgf.3	|- ( x = A -> ( ph <-> ps ) )
vtoclgf.4	|- ph

Assertion

Ref	Expression
vtoclgf	|- ( A e. V -> ps )

Proof of Theorem vtoclgf

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		vtoclgf.1	. . . 4 |- F/_ x A
3	2	issetf 3208	. . 3 |- ( A e. _V <-> E. x x = A )
4		vtoclgf.2	. . . 4 |- F/ x ps
5		vtoclgf.4	. . . . 5 |- ph
6		vtoclgf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	mpbii 223	. . . 4 |- ( x = A -> ps )
8	4, 7	exlimi 2086	. . 3 |- ( E. x x = A -> ps )
9	3, 8	sylbi 207	. 2 |- ( A e. _V -> ps )
10	1, 9	syl 17	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  vtocl2gf  3268  vtocl3gf  3269  vtoclgaf  3271  elabgf  3348  fprodsplit1f  14721  ssiun2sf  29378  subtr  32308  subtr2  32309  supxrgere  39549  supxrgelem  39553  supxrge  39554  fsumsplit1  39804  fmuldfeqlem1  39814  fprodcnlem  39831  climsuse  39840  dvnmptdivc  40153  dvmptfprodlem  40159  stoweidlem59  40276  fourierdlem31  40355  sge0f1o  40599  sge0fodjrnlem  40633  salpreimagelt  40918  salpreimalegt  40920