Theorem elabgf1 16480

Description: One implication of elabgf 2949. (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 16479	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )
4		elabgf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
5	3, 4	mpg 1500	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1398   F/wnf 1509   e. wcel 2202   {cab 2217   F/_wnfc 2362

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  elabf1  16482