Theorem abeq2d 1564

Description: Equality of a class variable and a class abstraction (deduction).

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 1533	. 2 |- (ph -> (x e. A <-> x e. {x | ps}))
3		abid 1458	. 2 |- (x e. {x | ps} <-> ps)
4	2, 3	syl6bb 534	1 |- (ph -> (x e. A <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  {cab 1456

This theorem is referenced by:  genpn0 5078  genpss 5079  genpnmax 5082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465

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