Theorem elabf1 14760

Description: One implication of elabf 2892. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabf1.nf	|- F/ x ps
elabf1.1	|- ( x = A -> ( ph -> ps ) )

Assertion

Ref	Expression
elabf1	|- ( A e. { x | ph } -> ps )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem elabf1

Step	Hyp	Ref	Expression
1		nfcv 2329	. 2 |- F/_ x A
2		elabf1.nf	. 2 |- F/ x ps
3		elabf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
4	1, 2, 3	elabgf1 14758	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1363   F/wnf 1470   e. wcel 2158   {cab 2173

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751

This theorem is referenced by:  elab1  14762  bj-bdfindis  14926