Theorem rexdifpr 4205

Description: Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)

Assertion

Ref	Expression
rexdifpr	|- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )

Proof of Theorem rexdifpr

Step	Hyp	Ref	Expression
1		eldifpr 4204	. . . . 5 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ x =/= B /\ x =/= C ) )
2		3anass 1042	. . . . 5 |- ( ( x e. A /\ x =/= B /\ x =/= C ) <-> ( x e. A /\ ( x =/= B /\ x =/= C ) ) )
3	1, 2	bitri 264	. . . 4 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ ( x =/= B /\ x =/= C ) ) )
4	3	anbi1i 731	. . 3 |- ( ( x e. ( A \ { B , C } ) /\ ph ) <-> ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) )
5		anass 681	. . . 4 |- ( ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) <-> ( x e. A /\ ( ( x =/= B /\ x =/= C ) /\ ph ) ) )
6		df-3an 1039	. . . . . 6 |- ( ( x =/= B /\ x =/= C /\ ph ) <-> ( ( x =/= B /\ x =/= C ) /\ ph ) )
7	6	bicomi 214	. . . . 5 |- ( ( ( x =/= B /\ x =/= C ) /\ ph ) <-> ( x =/= B /\ x =/= C /\ ph ) )
8	7	anbi2i 730	. . . 4 |- ( ( x e. A /\ ( ( x =/= B /\ x =/= C ) /\ ph ) ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
9	5, 8	bitri 264	. . 3 |- ( ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
10	4, 9	bitri 264	. 2 |- ( ( x e. ( A \ { B , C } ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
11	10	rexbii2 3039	1 |- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {cpr 4179

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-3an 1039  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-ne 2795  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  usgr2pth0  26661