Theorem elabf1 14917

Description: One implication of elabf 2895. (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 2332	. 2 |- F/_ x A
2		elabf1.nf	. 2 |- F/ x ps
3		elabf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
4	1, 2, 3	elabgf1 14915	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1364   F/wnf 1471   e. wcel 2160   {cab 2175

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  elab1  14919  bj-bdfindis  15083