Theorem elabf2 13160

Description: One implication of elabf 2831. (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 2282	. . 3 |- F/_ x A
3		elabf2.nf	. . 3 |- F/ x ps
4		elabf2.1	. . 3 |- ( x = A -> ( ps -> ph ) )
5	2, 3, 4	elabgf2 13158	. 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 1332   F/wnf 1437   e. wcel 1481   {cab 2126   _Vcvv 2689

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691

This theorem is referenced by:  elab2a  13162  bj-bdfindis  13316