Theorem elabg2 15683

Description: One implication of elabg 2918. (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 2347	. 2 |- F/_ x A
2		nfv 1550	. 2 |- F/ x ps
3		elabg2.1	. 2 |- ( x = A -> ( ps -> ph ) )
4	1, 2, 3	elabgf2 15678	1 |- ( A e. V -> ( ps -> A e. { x | ph } ) )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   {cab 2190

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773

This theorem is referenced by: (None)