Theorem rabbi2dva 3335

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 3128	. 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 2718	. 2 |- ( ph -> { x e. A | x e. B } = { x e. A | ps } )
4	1, 3	eqtrid 2215	1 |- ( ph -> ( A i^i B ) = { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {crab 2452   i^i cin 3120

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-ral 2453  df-rab 2457  df-in 3127

This theorem is referenced by:  fndmdif  5601  txcnmpt  13067