Theorem cleq2ab 1179

Description: Equality of two class abstractions means their wff's are equivalent.

Assertion

Ref	Expression
cleq2ab	|- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))

Proof of Theorem cleq2ab

Step	Hyp	Ref	Expression
1		hbab1 1095	. . 3 |- (y e. {x | ph} -> A.x y e. {x | ph})
2		hbab1 1095	. . 3 |- (y e. {x | ps} -> A.x y e. {x | ps})
3	1, 2	cleqf 1167	. 2 |- ({x | ph} = {x | ps} <-> A.x(x e. {x | ph} <-> x e. {x | ps}))
4		abid 1094	. . . 4 |- (x e. {x | ph} <-> ph)
5		abid 1094	. . . 4 |- (x e. {x | ps} <-> ps)
6	4, 5	bibi12i 462	. . 3 |- ((x e. {x | ph} <-> x e. {x | ps}) <-> (ph <-> ps))
7	6	bial 695	. 2 |- (A.x(x e. {x | ph} <-> x e. {x | ps}) <-> A.x(ph <-> ps))
8	3, 7	bitr 151	1 |- ({x | ph} = {x | ps} <-> A.x(ph <-> ps))

Syntax hints:   <-> wb 127  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092

This theorem is referenced by:  biabi 1181  biabd 1182  pw2en 3348  karden 3551

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-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  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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