Theorem abbi2dv 2326

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 1898	. 2 |- ( ph -> A. x ( x e. A <-> ps ) )
3		abeq2 2316	. 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 1371   = wceq 1373   e. wcel 2178   {cab 2193

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203

This theorem is referenced by:  sbab  2335  iftrue  3584  iffalse  3587  iniseg  5073  fncnvima2  5724  isoini  5910  dftpos3  6371  unfiexmid  7041  tgval3  14645  txrest  14863  cnblcld  15122