Theorem birabsdv 1344

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule).

Hypothesis

Ref	Expression
birabsdv.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
birabsdv	|- (ph -> {x e. A | ps} = {x e. A | ch})

Distinct variable group(s):   ph,x

Proof of Theorem birabsdv

Step	Hyp	Ref	Expression
1		birabsdv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	a1d 14	. 2 |- (ph -> (x e. A -> (ps <-> ch)))
3	2	birabdv 1343	1 |- (ph -> {x e. A | ps} = {x e. A | ch})

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092  {crab 1204

This theorem is referenced by:  supeq1 2155  inf3lema 3460  tz9.12lem1 3503  tz9.12lem3 3505  rankval 3512  rankvalg 3513  rankonid 3538  zornlem1 3603  zornlem6 3608  zornlem7 3609  zorn 3611  oncardval 3626  cardval 3633  alephon 3671  alephsuc 3672  alephsuc2 3686  subval 4134  divval 4217  flvalt 4623  sqrval 4729  revalt 4794  imvalt 4795  ocvalt 5161  pjvalt 5246  shsumvalt 5279  spanvalt 5300  hsupval2t 5301  chsupid 5312

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  df-rab 1208

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