Theorem vtoclgf 3165

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 3118	. 2 |- ( A e. V -> A e. _V )
2		vtoclgf.1	. . . 4 |- F/_ x A
3	2	issetf 3114	. . 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 211	. . . 4 |- ( x = A -> ps )
8	4, 7	exlimi 1913	. . 3 |- ( E. x x = A -> ps )
9	3, 8	sylbi 195	. 2 |- ( A e. _V -> ps )
10	1, 9	syl 16	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1395   E.wex 1613   F/wnf 1617   e. wcel 1819   F/_wnfc 2605   _Vcvv 3109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111

This theorem is referenced by:  vtocl2gf  3169  vtocl3gf  3170  vtoclgaf  3172  elabgf  3244  ssiun2sf  27565  subtr  30337  subtr2  30338  fsumsplit1  31776  fmuldfeqlem1  31779  fprodsplit1f  31796  climsuse  31817  dvnmptdivc  31938  dvmptfprodlem  31944  stoweidlem59  32044  fourierdlem31  32123