Theorem elabg2 11989

Description: One implication of elabg 2764. (Contributed by BJ, 21-Nov-2019.)

Hypothesis

Ref	Expression
elabg2.1	|- ( x = A -> ( ps -> ph ) )

Assertion

Ref	Expression
elabg2	|- ( A e. V -> ( ps -> A e. { x | ph } ) )

Distinct variable groups:   ps, x   x, A

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

Proof of Theorem elabg2

Step	Hyp	Ref	Expression
1		nfcv 2229	. 2 |- F/_ x A
2		nfv 1467	. 2 |- F/ x ps
3		elabg2.1	. 2 |- ( x = A -> ( ps -> ph ) )
4	1, 2, 3	elabgf2 11984	1 |- ( A e. V -> ( ps -> A e. { x | ph } ) )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439   {cab 2075

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624

This theorem is referenced by: (None)