Theorem raldifb 3303

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 2463	. . . . . 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 3166	. . . . . 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 2507	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 2167   e/ wnel 2462   A.wral 2475   \ cdif 3154

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-nel 2463  df-ral 2480  df-v 2765  df-dif 3159

This theorem is referenced by: (None)