Theorem abbi2dv 2009

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 1664	. 2 |- (ph -> A.x(x e. A <-> ps))
3		abeq2 1999	. 2 |- (A = {x | ps} <-> A.x(x e. A <-> ps))
4	2, 3	sylibr 217	1 |- (ph -> A = {x | ps})

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871

This theorem is referenced by:  sbab 2015  rabbirdvOLD 2802  iftrue 2989  iftrueOLD 2990  iffalse 2991  iin0 3477  iniseg 4296  isoini 4877  pw2en 5505  r1val2 5789  aceq3 5895  tgval3 8895  metnei 9155  metcls 9244  grpinvf 9364  fictb 15371

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880

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