Theorem elabf2 13663

Description: One implication of elabf 2869. (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 2308	. . 3 |- F/_ x A
3		elabf2.nf	. . 3 |- F/ x ps
4		elabf2.1	. . 3 |- ( x = A -> ( ps -> ph ) )
5	2, 3, 4	elabgf2 13661	. 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 1343   F/wnf 1448   e. wcel 2136   {cab 2151   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  elab2a  13665  bj-bdfindis  13829