Theorem elabf2 16724

Description: One implication of elabf 2969. (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 2392	. . 3 |- F/_ x A
3		elabf2.nf	. . 3 |- F/ x ps
4		elabf2.1	. . 3 |- ( x = A -> ( ps -> ph ) )
5	2, 3, 4	elabgf2 16722	. 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 1402   F/wnf 1513   e. wcel 2209   {cab 2224   _Vcvv 2821

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  elab2a  16726  bj-bdfindis  16887