Theorem raldifb 3211

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 261	. . . 4 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. A -> ( x e/ B -> ph ) ) )
2	1	bicomi 131	. . 3 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( ( x e. A /\ x e/ B ) -> ph ) )
3		df-nel 2402	. . . . . 6 |- ( x e/ B <-> -. x e. B )
4	3	anbi2i 452	. . . . 5 |- ( ( x e. A /\ x e/ B ) <-> ( x e. A /\ -. x e. B ) )
5		eldif 3075	. . . . . 6 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
6	5	bicomi 131	. . . . 5 |- ( ( x e. A /\ -. x e. B ) <-> x e. ( A \ B ) )
7	4, 6	bitri 183	. . . 4 |- ( ( x e. A /\ x e/ B ) <-> x e. ( A \ B ) )
8	7	imbi1i 237	. . 3 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. ( A \ B ) -> ph ) )
9	2, 8	bitri 183	. 2 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( x e. ( A \ B ) -> ph ) )
10	9	ralbii2 2443	1 |- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   e/ wnel 2401   A.wral 2414   \ cdif 3063

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-nel 2402  df-ral 2419  df-v 2683  df-dif 3068

This theorem is referenced by: (None)