Theorem bj-vtoclgft 16547

Description: Weakening two hypotheses of vtoclgf 2873. (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 2825	. 2 |- ( A e. V -> A e. _V )
2		bj-vtoclgf.nf1	. . . 4 |- F/_ x A
3	2	issetf 2821	. . 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 16541	. . 3 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> ps ) )
7	3, 6	biimtrid 152	. 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 1396   = wceq 1398   F/wnf 1509   E.wex 1541   e. wcel 2203   F/_wnfc 2371   _Vcvv 2813

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815

This theorem is referenced by:  bj-vtoclgf  16548  elabgft1  16550  bj-rspgt  16558