Theorem rexdifpr 3632

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 3631	. . . . 5 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ x =/= B /\ x =/= C ) )
2		3anass 983	. . . . 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 981	. . . . . 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 2498	1 |- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 979   e. wcel 2158   =/= wne 2357   E.wrex 2466   \ cdif 3138   {cpr 3605

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-sn 3610  df-pr 3611

This theorem is referenced by: (None)