Theorem abbi2dv 2353

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 1923	. 2 |- ( ph -> A. x ( x e. A <-> ps ) )
3		abeq2 2341	. 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 1396   = wceq 1398   e. wcel 2203   {cab 2218

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228

This theorem is referenced by:  sbab  2362  iftrue  3627  iffalse  3630  iniseg  5134  fncnvima2  5799  isoini  5991  dftpos3  6493  unfiexmid  7178  tgval3  14923  txrest  15141  cnblcld  15400