Theorem abbi2dv 1554

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 1268	. 2 |- (ph -> A.x(x e. A <-> ps))
3		abeq2 1544	. 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 950   = wceq 1099   e. wcel 1105  {cab 1440

This theorem is referenced by:  sbab 1559  rabbirdv 2192  iftrue 2337  iffalse 2338  iin0 2708  iniseg 3381  isoini 3839  pw2en 4380  r1val2 4602  aceq3 4657  tgval3t 7518  grpinvf 7962

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449

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