Theorem raldifb 3313

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 263	. . . 4 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. A -> ( x e/ B -> ph ) ) )
2	1	bicomi 132	. . 3 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( ( x e. A /\ x e/ B ) -> ph ) )
3		df-nel 2472	. . . . . 6 |- ( x e/ B <-> -. x e. B )
4	3	anbi2i 457	. . . . 5 |- ( ( x e. A /\ x e/ B ) <-> ( x e. A /\ -. x e. B ) )
5		eldif 3175	. . . . . 6 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
6	5	bicomi 132	. . . . 5 |- ( ( x e. A /\ -. x e. B ) <-> x e. ( A \ B ) )
7	4, 6	bitri 184	. . . 4 |- ( ( x e. A /\ x e/ B ) <-> x e. ( A \ B ) )
8	7	imbi1i 238	. . 3 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. ( A \ B ) -> ph ) )
9	2, 8	bitri 184	. 2 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( x e. ( A \ B ) -> ph ) )
10	9	ralbii2 2516	1 |- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   e/ wnel 2471   A.wral 2484   \ cdif 3163

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-nel 2472  df-ral 2489  df-v 2774  df-dif 3168

This theorem is referenced by: (None)