Theorem elabgf0 13812

Description: Lemma for elabgf 2872. (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		eleq1 2233	. 2 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
2		abid 2158	. 2 |- ( x e. { x | ph } <-> ph )
3	1, 2	bitr3di 194	1 |- ( x = A -> ( A e. { x | ph } <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166

This theorem is referenced by:  elabgft1  13813  elabgf2  13815