Theorem eqabrd 2370

Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2368). (Contributed by NM, 16-Nov-1995.)

Hypothesis

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

Assertion

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

Proof of Theorem eqabrd

Step	Hyp	Ref	Expression
1		eqabrd.1	. . 3 |- ( ph -> A = { x | ps } )
2	1	eleq2d 2302	. 2 |- ( ph -> ( x e. A <-> x e. { x | ps } ) )
3		abid 2220	. 2 |- ( x e. { x | ps } <-> ps )
4	2, 3	bitrdi 196	1 |- ( ph -> ( x e. A <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228

This theorem is referenced by:  mapsnend  7043