Theorem elabf1 9920

Description: One implication of elabf 2686. (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 2178	. 2 |- F/_ x A
2		elabf1.nf	. 2 |- F/ x ps
3		elabf1.1	. 2 |- ( x = A -> ( ph -> ps ) )
4	1, 2, 3	elabgf1 9918	1 |- ( A e. { x | ph } -> ps )

Syntax hints:   -> wi 4   = wceq 1243   F/wnf 1349   e. wcel 1393   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  elab1  9922  bj-bdfindis  10072