Theorem elabgf1 13814

Description: One implication of elabgf 2872. (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 13813	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )
4		elabgf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
5	3, 4	mpg 1444	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1348   F/wnf 1453   e. wcel 2141   {cab 2156   F/_wnfc 2299

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  elabf1  13816