Theorem elabf2 15918

Description: One implication of elabf 2923. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabf2.nf	|- F/ x ps
elabf2.s	|- A e. _V
elabf2.1	|- ( x = A -> ( ps -> ph ) )

Assertion

Ref	Expression
elabf2	|- ( ps -> A e. { x | ph } )

Distinct variable group:   x, A

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

Proof of Theorem elabf2

Step	Hyp	Ref	Expression
1		elabf2.s	. 2 |- A e. _V
2		nfcv 2350	. . 3 |- F/_ x A
3		elabf2.nf	. . 3 |- F/ x ps
4		elabf2.1	. . 3 |- ( x = A -> ( ps -> ph ) )
5	2, 3, 4	elabgf2 15916	. 2 |- ( A e. _V -> ( ps -> A e. { x | ph } ) )
6	1, 5	ax-mp 5	1 |- ( ps -> A e. { x | ph } )

Syntax hints:   -> wi 4   = wceq 1373   F/wnf 1484   e. wcel 2178   {cab 2193   _Vcvv 2776

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  elab2a  15920  bj-bdfindis  16082