Theorem elabgf1 13660

Description: One implication of elabgf 2868. (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 13659	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )
4		elabgf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
5	3, 4	mpg 1439	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1343   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295

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:  elabf1  13662