Theorem vtoclgf 2657

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 2610	. 2 |- ( A e. V -> A e. _V )
2		vtoclgf.1	. . . 4 |- F/_ x A
3	2	issetf 2606	. . 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 146	. . . 4 |- ( x = A -> ps )
8	4, 7	exlimi 1525	. . 3 |- ( E. x x = A -> ps )
9	3, 8	sylbi 119	. 2 |- ( A e. _V -> ps )
10	1, 9	syl 14	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433   F/_wnfc 2206   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  vtoclg  2658  vtocl2gf  2660  vtocl3gf  2661  vtoclgaf  2663  ceqsexg  2723  elabgf  2736  mob  2774  opeliunxp2  4494  fvmptss2  5268