Theorem vtoclgf 3174

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 3127	. 2 |- ( A e. V -> A e. _V )
2		vtoclgf.1	. . . 4 |- F/_ x A
3	2	issetf 3123	. . 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 1859	. . 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 1379   E.wex 1596   F/wnf 1599   e. wcel 1767   F/_wnfc 2615   _Vcvv 3118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120

This theorem is referenced by:  vtocl2gf  3178  vtocl3gf  3179  vtoclgaf  3181  elabgf  3253  ssiun2sf  27241  subtr  30050  subtr2  30051  fmuldfeqlem1  31446  climsuse  31464  stoweidlem59  31673  fourierdlem31  31752  fourierdlem86  31807  fourierdlem89  31810  fourierdlem91  31812