Theorem abeq2d 2344

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 2301	. 2 |- ( ph -> ( x e. A <-> x e. { x | ps } ) )
3		abid 2219	. 2 |- ( x e. { x | ps } <-> ps )
4	2, 3	bitrdi 196	1 |- ( ph -> ( x e. A <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227

This theorem is referenced by:  fvelimab  5702  frecsuclem  6571