Theorem abeq2d 2463

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 2420	. 2 |- (ph -> (x e. A <-> x e. {x | ps}))
3		abid 2341	. 2 |- (x e. {x | ps} <-> ps)
4	2, 3	syl6bb 252	1 |- (ph -> (x e. A <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by:  fvelimab  5371