Theorem biabrdv 1184

Description: Deduction from a wff to a class abstraction.

Hypothesis

Ref	Expression
biabrdv.1	|- (ph -> (x e. A <-> ps))

Assertion

Ref	Expression
biabrdv	|- (ph -> A = {x | ps})

Distinct variable group(s):   x,A   ph,x

Proof of Theorem biabrdv

Step	Hyp	Ref	Expression
1		biabrdv.1	. . 3 |- (ph -> (x e. A <-> ps))
2	1	19.21aiv 943	. 2 |- (ph -> A.x(x e. A <-> ps))
3		cleqab 1174	. 2 |- (A = {x | ps} <-> A.x(x e. A <-> ps))
4	2, 3	sylibr 175	1 |- (ph -> A = {x | ps})

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092

This theorem is referenced by:  sbab 1188  birabrdv 1648  iftrue 1780  iffalse 1781  iniseg 2619  isoini 2938  pw2en 3348  r1val2 3522  aceq3 3556

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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