Theorem elabgf0 15274

Description: Lemma for elabgf 2902. (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 2256	. 2 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
2		abid 2181	. 2 |- ( x e. { x | ph } <-> ph )
3	1, 2	bitr3di 195	1 |- ( x = A -> ( A e. { x | ph } <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179

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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189

This theorem is referenced by:  elabgft1  15275  elabgf2  15277