Theorem elabgf0 12973

Description: Lemma for elabgf 2821. (Contributed by BJ, 21-Nov-2019.)

Assertion

Ref	Expression
elabgf0	|- ( x = A -> ( A e. { x | ph } <-> ph ) )

Proof of Theorem elabgf0

Step	Hyp	Ref	Expression
1		abid 2125	. 2 |- ( x e. { x | ph } <-> ph )
2		eleq1 2200	. 2 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
3	1, 2	syl5rbbr 194	1 |- ( x = A -> ( A e. { x | ph } <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2123

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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133

This theorem is referenced by:  elabgft1  12974  elabgf2  12976