Theorem raldifb 3083

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 250	. . . 4 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. A -> ( x e/ B -> ph ) ) )
2	1	bicomi 123	. . 3 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( ( x e. A /\ x e/ B ) -> ph ) )
3		df-nel 2207	. . . . . 6 |- ( x e/ B <-> -. x e. B )
4	3	anbi2i 430	. . . . 5 |- ( ( x e. A /\ x e/ B ) <-> ( x e. A /\ -. x e. B ) )
5		eldif 2927	. . . . . 6 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
6	5	bicomi 123	. . . . 5 |- ( ( x e. A /\ -. x e. B ) <-> x e. ( A \ B ) )
7	4, 6	bitri 173	. . . 4 |- ( ( x e. A /\ x e/ B ) <-> x e. ( A \ B ) )
8	7	imbi1i 227	. . 3 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. ( A \ B ) -> ph ) )
9	2, 8	bitri 173	. 2 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( x e. ( A \ B ) -> ph ) )
10	9	ralbii2 2334	1 |- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   e/ wnel 2205   A.wral 2306   \ cdif 2914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207  df-ral 2311  df-v 2559  df-dif 2920

This theorem is referenced by: (None)