Theorem raldifb 3112

Description: Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)

Assertion

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

Proof of Theorem raldifb

Step	Hyp	Ref	Expression
1		impexp 259	. . . 4 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. A -> ( x e/ B -> ph ) ) )
2	1	bicomi 130	. . 3 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( ( x e. A /\ x e/ B ) -> ph ) )
3		df-nel 2340	. . . . . 6 |- ( x e/ B <-> -. x e. B )
4	3	anbi2i 444	. . . . 5 |- ( ( x e. A /\ x e/ B ) <-> ( x e. A /\ -. x e. B ) )
5		eldif 2982	. . . . . 6 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
6	5	bicomi 130	. . . . 5 |- ( ( x e. A /\ -. x e. B ) <-> x e. ( A \ B ) )
7	4, 6	bitri 182	. . . 4 |- ( ( x e. A /\ x e/ B ) <-> x e. ( A \ B ) )
8	7	imbi1i 236	. . 3 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. ( A \ B ) -> ph ) )
9	2, 8	bitri 182	. 2 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( x e. ( A \ B ) -> ph ) )
10	9	ralbii2 2376	1 |- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   e/ wnel 2339   A.wral 2348   \ cdif 2970

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-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-nel 2340  df-ral 2353  df-v 2603  df-dif 2975

This theorem is referenced by: (None)