Theorem rabbi2dva 3345

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 3138	. 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 2727	. 2 |- ( ph -> { x e. A | x e. B } = { x e. A | ps } )
4	1, 3	eqtrid 2222	1 |- ( ph -> ( A i^i B ) = { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   {crab 2459   i^i cin 3130

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-ral 2460  df-rab 2464  df-in 3137

This theorem is referenced by:  fndmdif  5623  txcnmpt  13858