Theorem rexdifpr 3650

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 3649	. . . . 5 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ x =/= B /\ x =/= C ) )
2		3anass 984	. . . . 5 |- ( ( x e. A /\ x =/= B /\ x =/= C ) <-> ( x e. A /\ ( x =/= B /\ x =/= C ) ) )
3	1, 2	bitri 184	. . . 4 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ ( x =/= B /\ x =/= C ) ) )
4	3	anbi1i 458	. . 3 |- ( ( x e. ( A \ { B , C } ) /\ ph ) <-> ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) )
5		anass 401	. . . 4 |- ( ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) <-> ( x e. A /\ ( ( x =/= B /\ x =/= C ) /\ ph ) ) )
6		df-3an 982	. . . . . 6 |- ( ( x =/= B /\ x =/= C /\ ph ) <-> ( ( x =/= B /\ x =/= C ) /\ ph ) )
7	6	bicomi 132	. . . . 5 |- ( ( ( x =/= B /\ x =/= C ) /\ ph ) <-> ( x =/= B /\ x =/= C /\ ph ) )
8	7	anbi2i 457	. . . 4 |- ( ( x e. A /\ ( ( x =/= B /\ x =/= C ) /\ ph ) ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
9	5, 8	bitri 184	. . 3 |- ( ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
10	4, 9	bitri 184	. 2 |- ( ( x e. ( A \ { B , C } ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
11	10	rexbii2 2508	1 |- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2167   =/= wne 2367   E.wrex 2476   \ cdif 3154   {cpr 3623

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-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-sn 3628  df-pr 3629

This theorem is referenced by: (None)