Theorem abeq2d 2734

Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2732). (Contributed by NM, 16-Nov-1995.)

Hypothesis

Ref	Expression
abeq2d.1	|- ( ph -> A = { x | ps } )

Assertion

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

Proof of Theorem abeq2d

Step	Hyp	Ref	Expression
1		abeq2d.1	. . 3 |- ( ph -> A = { x | ps } )
2	1	eleq2d 2687	. 2 |- ( ph -> ( x e. A <-> x e. { x | ps } ) )
3		abid 2610	. 2 |- ( x e. { x | ps } <-> ps )
4	2, 3	syl6bb 276	1 |- ( ph -> ( x e. A <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by:  abeq2i  2735  fvelimab  6253  nosupbnd2  31862  ispridlc  33869  ac6s6  33980  dib1dim  36454  mapsnend  39391