Theorem rabbi2dva 3821

Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)

Hypothesis

Ref	Expression
rabbi2dva.1	|- ( ( ph /\ x e. A ) -> ( x e. B <-> ps ) )

Assertion

Ref	Expression
rabbi2dva	|- ( ph -> ( A i^i B ) = { x e. A | ps } )

Distinct variable groups:   ph, x   x, A   x, B

Allowed substitution hint:   ps( x)

Proof of Theorem rabbi2dva

Step	Hyp	Ref	Expression
1		dfin5 3582	. 2 |- ( A i^i B ) = { x e. A | x e. B }
2		rabbi2dva.1	. . 3 |- ( ( ph /\ x e. A ) -> ( x e. B <-> ps ) )
3	2	rabbidva 3188	. 2 |- ( ph -> { x e. A | x e. B } = { x e. A | ps } )
4	1, 3	syl5eq 2668	1 |- ( ph -> ( A i^i B ) = { x e. A | ps } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rab 2921  df-in 3581

This theorem is referenced by:  fndmdif  6321  bitsshft  15197  sylow3lem2  18043  leordtvallem1  21014  leordtvallem2  21015  ordtt1  21183  xkoccn  21422  txcnmpt  21427  xkopt  21458  ordthmeolem  21604  qustgphaus  21926  itg2monolem1  23517  lhop1  23777  efopn  24404  dirith  25218  pjvec  28555  pjocvec  28556  neibastop3  32357  diarnN  36418