Theorem elabgf1 13157

Description: One implication of elabgf 2830. (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 13156	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )
4		elabgf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
5	3, 4	mpg 1428	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1332   F/wnf 1437   e. wcel 1481   {cab 2126   F/_wnfc 2269

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:  elabf1  13159