Theorem rexdifpr 3661

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 3660	. . . . 5 |- ( x e. ( A \ { B , C } ) <-> ( x e. A /\ x =/= B /\ x =/= C ) )
2		3anass 985	. . . . 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 983	. . . . . 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 2517	1 |- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 981   e. wcel 2176   =/= wne 2376   E.wrex 2485   \ cdif 3163   {cpr 3634

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by: (None)