Theorem cleqabd 1178

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

Hypothesis

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

Assertion

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

Proof of Theorem cleqabd

Step	Hyp	Ref	Expression
1		cleqabd.1	. . 3 |- (ph -> A = {x | ps})
2	1	eleq2d 1156	. 2 |- (ph -> (x e. A <-> x e. {x | ps}))
3		abid 1094	. 2 |- (x e. {x | ps} <-> ps)
4	2, 3	syl6bb 414	1 |- (ph -> (x e. A <-> ps))

Syntax hints:   -> wi 2   <-> wb 127  {cab 1090   = wceq 1091   e. wcel 1092

This theorem is referenced by:  genpn0 3900  genpss 3901  genpnmax 3904

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

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