Theorem bj-vtoclgft 13153

Description: Weakening two hypotheses of vtoclgf 2747. (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 2700	. 2 |- ( A e. V -> A e. _V )
2		bj-vtoclgf.nf1	. . . 4 |- F/_ x A
3	2	issetf 2696	. . 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 13147	. . 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 1330   = wceq 1332   F/wnf 1437   E.wex 1469   e. wcel 1481   F/_wnfc 2269   _Vcvv 2689

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691

This theorem is referenced by:  bj-vtoclgf  13154  elabgft1  13156  bj-rspgt  13164