Theorem rexdifsn 3655

Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)

Assertion

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

Proof of Theorem rexdifsn

Step	Hyp	Ref	Expression
1		eldifsn 3650	. . . 4 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2	1	anbi1i 453	. . 3 |- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( ( x e. A /\ x =/= B ) /\ ph ) )
3		anass 398	. . 3 |- ( ( ( x e. A /\ x =/= B ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
4	2, 3	bitri 183	. 2 |- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
5	4	rexbii2 2446	1 |- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1480   =/= wne 2308   E.wrex 2417   \ cdif 3068   {csn 3527

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-rex 2422  df-v 2688  df-dif 3073  df-sn 3533

This theorem is referenced by: (None)