Theorem abbi2dv 2233

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

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

Allowed substitution hint:   ps( x)

Proof of Theorem abbi2dv

Step	Hyp	Ref	Expression
1		abbirdv.1	. . 3 |- ( ph -> ( x e. A <-> ps ) )
2	1	alrimiv 1828	. 2 |- ( ph -> A. x ( x e. A <-> ps ) )
3		abeq2 2223	. 2 |- ( A = { x | ps } <-> A. x ( x e. A <-> ps ) )
4	2, 3	sylibr 133	1 |- ( ph -> A = { x | ps } )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1312   = wceq 1314   e. wcel 1463   {cab 2101

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111

This theorem is referenced by:  sbab  2241  iftrue  3445  iffalse  3448  iniseg  4869  fncnvima2  5495  isoini  5673  dftpos3  6113  unfiexmid  6759  tgval3  12070  txrest  12287  cnblcld  12524