Theorem bj-vtoclgft 13656

Description: Weakening two hypotheses of vtoclgf 2784. (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 2737	. 2 |- ( A e. V -> A e. _V )
2		bj-vtoclgf.nf1	. . . 4 |- F/_ x A
3	2	issetf 2733	. . 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 13650	. . 3 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( E. x x = A -> ps ) )
7	3, 6	syl5bi 151	. 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 1341   = wceq 1343   F/wnf 1448   E.wex 1480   e. wcel 2136   F/_wnfc 2295   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  bj-vtoclgf  13657  elabgft1  13659  bj-rspgt  13667