Theorem bj-vtoclgf 16518

Description: Weakening two hypotheses of vtoclgf 2872. (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 )
bj-vtoclgf.maj	|- ( x = A -> ( ph -> ps ) )

Assertion

Ref	Expression
bj-vtoclgf	|- ( A e. V -> ps )

Proof of Theorem bj-vtoclgf

Step	Hyp	Ref	Expression
1		bj-vtoclgf.nf1	. . 3 |- F/_ x A
2		bj-vtoclgf.nf2	. . 3 |- F/ x ps
3		bj-vtoclgf.min	. . 3 |- ( x = A -> ph )
4	1, 2, 3	bj-vtoclgft 16517	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ps ) )
5		bj-vtoclgf.maj	. 2 |- ( x = A -> ( ph -> ps ) )
6	4, 5	mpg 1500	1 |- ( A e. V -> ps )

Syntax hints:   -> wi 4   = wceq 1398   F/wnf 1509   e. wcel 2203   F/_wnfc 2371

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 2814

This theorem is referenced by:  elabgf2  16522