Theorem rabbi2dva 3412

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 3204	. 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 2787	. 2 |- ( ph -> { x e. A | x e. B } = { x e. A | ps } )
4	1, 3	eqtrid 2274	1 |- ( ph -> ( A i^i B ) = { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   i^i cin 3196

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-ral 2513  df-rab 2517  df-in 3203

This theorem is referenced by:  fndmdif  5739  txcnmpt  14941