Theorem abeq2d 2279

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 2236	. 2 |- ( ph -> ( x e. A <-> x e. { x | ps } ) )
3		abid 2153	. 2 |- ( x e. { x | ps } <-> ps )
4	2, 3	bitrdi 195	1 |- ( ph -> ( x e. A <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161

This theorem is referenced by:  fvelimab  5542  frecsuclem  6374