Theorem rabbi2dva 3197

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 2995	. 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 2603	. 2 |- ( ph -> { x e. A | x e. B } = { x e. A | ps } )
4	1, 3	syl5eq 2129	1 |- ( ph -> ( A i^i B ) = { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1287   e. wcel 1436   {crab 2359   i^i cin 2987

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-ral 2360  df-rab 2364  df-in 2994

This theorem is referenced by:  fndmdif  5361