Theorem elabf1 16103

Description: One implication of elabf 2946. (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 2372	. 2 |- F/_ x A
2		elabf1.nf	. 2 |- F/ x ps
3		elabf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
4	1, 2, 3	elabgf1 16101	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1395   F/wnf 1506   e. wcel 2200   {cab 2215

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  elab1  16105  bj-bdfindis  16268