Theorem abbi2dv 2296

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 1874	. 2 |- ( ph -> A. x ( x e. A <-> ps ) )
3		abeq2 2286	. 2 |- ( A = { x | ps } <-> A. x ( x e. A <-> ps ) )
4	2, 3	sylibr 134	1 |- ( ph -> A = { x | ps } )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   {cab 2163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173

This theorem is referenced by:  sbab  2305  iftrue  3541  iffalse  3544  iniseg  5002  fncnvima2  5640  isoini  5822  dftpos3  6266  unfiexmid  6920  tgval3  13698  txrest  13916  cnblcld  14175