Theorem elabgf1 15915

Description: One implication of elabgf 2922. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabgf1.nf1	|- F/_ x A
elabgf1.nf2	|- F/ x ps
elabgf1.1	|- ( x = A -> ( ph -> ps ) )

Assertion

Ref	Expression
elabgf1	|- ( A e. { x | ph } -> ps )

Proof of Theorem elabgf1

Step	Hyp	Ref	Expression
1		elabgf1.nf1	. . 3 |- F/_ x A
2		elabgf1.nf2	. . 3 |- F/ x ps
3	1, 2	elabgft1 15914	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )
4		elabgf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
5	3, 4	mpg 1475	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1373   F/wnf 1484   e. wcel 2178   {cab 2193   F/_wnfc 2337

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  elabf1  15917