Theorem bj-vtoclgft 10585

Description: Weakening two hypotheses of vtoclgf 2657. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
bj-vtoclgf.nf1	|- F/_ x A
bj-vtoclgf.nf2	|- F/ x ps
bj-vtoclgf.min	|- ( x = A -> ph )

Assertion

Ref	Expression
bj-vtoclgft	|- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) )

Proof of Theorem bj-vtoclgft

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( A e. V -> A e. _V )
2		bj-vtoclgf.nf1	. . . 4 |- F/_ x A
3	2	issetf 2606	. . 3 |- ( A e. _V <-> E. x x = A )
4		bj-vtoclgf.nf2	. . . 4 |- F/ x ps
5		bj-vtoclgf.min	. . . 4 |- ( x = A -> ph )
6	4, 5	bj-exlimmp 10580	. . 3 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> ps ) )
7	3, 6	syl5bi 150	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. _V -> ps ) )
8	1, 7	syl5 32	1 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) )

Syntax hints:   -> wi 4   A.wal 1282   = 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:  bj-vtoclgf  10586  elabgft1  10588  bj-rspgt  10596