Theorem abbi2dv 2256

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 1846	. 2 |- ( ph -> A. x ( x e. A <-> ps ) )
3		abeq2 2246	. 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 1329   = wceq 1331   e. wcel 1480   {cab 2123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133

This theorem is referenced by:  sbab  2265  iftrue  3474  iffalse  3477  iniseg  4906  fncnvima2  5534  isoini  5712  dftpos3  6152  unfiexmid  6799  tgval3  12216  txrest  12434  cnblcld  12693