Theorem abeq2d 2200

Description: Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)

Hypothesis

Ref	Expression
abeqd.1	|- ( ph -> A = { x | ps } )

Assertion

Ref	Expression
abeq2d	|- ( ph -> ( x e. A <-> ps ) )

Proof of Theorem abeq2d

Step	Hyp	Ref	Expression
1		abeqd.1	. . 3 |- ( ph -> A = { x | ps } )
2	1	eleq2d 2157	. 2 |- ( ph -> ( x e. A <-> x e. { x | ps } ) )
3		abid 2076	. 2 |- ( x e. { x | ps } <-> ps )
4	2, 3	syl6bb 194	1 |- ( ph -> ( x e. A <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   e. wcel 1438   {cab 2074

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084

This theorem is referenced by:  fvelimab  5360  frecsuclem  6171