Theorem abeq2d 2318

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 2275	. 2 |- ( ph -> ( x e. A <-> x e. { x | ps } ) )
3		abid 2193	. 2 |- ( x e. { x | ps } <-> ps )
4	2, 3	bitrdi 196	1 |- ( ph -> ( x e. A <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   {cab 2191

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201

This theorem is referenced by:  fvelimab  5635  frecsuclem  6492