Theorem raldifb 3750

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 462	. . 3 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. A -> ( x e/ B -> ph ) ) )
2		df-nel 2898	. . . . . 6 |- ( x e/ B <-> -. x e. B )
3	2	anbi2i 730	. . . . 5 |- ( ( x e. A /\ x e/ B ) <-> ( x e. A /\ -. x e. B ) )
4		eldif 3584	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5	3, 4	bitr4i 267	. . . 4 |- ( ( x e. A /\ x e/ B ) <-> x e. ( A \ B ) )
6	5	imbi1i 339	. . 3 |- ( ( ( x e. A /\ x e/ B ) -> ph ) <-> ( x e. ( A \ B ) -> ph ) )
7	1, 6	bitr3i 266	. 2 |- ( ( x e. A -> ( x e/ B -> ph ) ) <-> ( x e. ( A \ B ) -> ph ) )
8	7	ralbii2 2978	1 |- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   e/ wnel 2897   A.wral 2912   \ cdif 3571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-v 3202  df-dif 3577

This theorem is referenced by:  raldifsnb  4325  coprmproddvdslem  15376  poimirlem26  33435  aacllem  42547