Theorem abbi2dv 1578

Description: Deduction from a wff to a class abstraction.

Hypothesis

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

Assertion

Ref	Expression
abbi2dv	|- (ph -> A = {x | ps})

Distinct variable groups:   x,A   ph,x

Proof of Theorem abbi2dv

Step	Hyp	Ref	Expression
1		abbirdv.1	. . 3 |- (ph -> (x e. A <-> ps))
2	1	19.21aiv 1286	. 2 |- (ph -> A.x(x e. A <-> ps))
3		abeq2 1568	. 2 |- (A = {x | ps} <-> A.x(x e. A <-> ps))
4	2, 3	sylibr 200	1 |- (ph -> A = {x | ps})

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  {cab 1463

This theorem is referenced by:  sbab 1583  rabbirdv 2221  iftrue 2366  iffalse 2367  iin0 2740  iniseg 3430  isoini 3900  pw2en 4446  r1val2 4678  aceq3 4733  tgval3t 7625  metnei 7878  metcls 7966  grpinvf 8079

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472

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