Theorem vtoclgf 2810

Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of distinct 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 2763	. 2 |- ( A e. V -> A e. _V )
2		vtoclgf.1	. . . 4 |- F/_ x A
3	2	issetf 2759	. . 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 148	. . . 4 |- ( x = A -> ps )
8	4, 7	exlimi 1605	. . 3 |- ( E. x x = A -> ps )
9	3, 8	sylbi 121	. 2 |- ( A e. _V -> ps )
10	1, 9	syl 14	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   F/wnf 1471   E.wex 1503   e. wcel 2160   F/_wnfc 2319   _Vcvv 2752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  vtoclg  2812  vtocl2gf  2814  vtocl3gf  2815  vtoclgaf  2817  ceqsexg  2880  elabgf  2894  mob  2934  opeliunxp2  4785  fvmptss2  5612