Theorem rexdifpr 3584

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 3583	. . . . 5 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ x =/= B /\ x =/= C ) )
2		3anass 967	. . . . 5 |- ( ( x e. A /\ x =/= B /\ x =/= C ) <-> ( x e. A /\ ( x =/= B /\ x =/= C ) ) )
3	1, 2	bitri 183	. . . 4 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ ( x =/= B /\ x =/= C ) ) )
4	3	anbi1i 454	. . 3 |- ( ( x e. ( A \ { B , C } ) /\ ph ) <-> ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) )
5		anass 399	. . . 4 |- ( ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) <-> ( x e. A /\ ( ( x =/= B /\ x =/= C ) /\ ph ) ) )
6		df-3an 965	. . . . . 6 |- ( ( x =/= B /\ x =/= C /\ ph ) <-> ( ( x =/= B /\ x =/= C ) /\ ph ) )
7	6	bicomi 131	. . . . 5 |- ( ( ( x =/= B /\ x =/= C ) /\ ph ) <-> ( x =/= B /\ x =/= C /\ ph ) )
8	7	anbi2i 453	. . . 4 |- ( ( x e. A /\ ( ( x =/= B /\ x =/= C ) /\ ph ) ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
9	5, 8	bitri 183	. . 3 |- ( ( ( x e. A /\ ( x =/= B /\ x =/= C ) ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
10	4, 9	bitri 183	. 2 |- ( ( x e. ( A \ { B , C } ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ x =/= C /\ ph ) ) )
11	10	rexbii2 2465	1 |- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 963   e. wcel 2125   =/= wne 2324   E.wrex 2433   \ cdif 3095   {cpr 3557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-rex 2438  df-v 2711  df-dif 3100  df-un 3102  df-sn 3562  df-pr 3563

This theorem is referenced by: (None)