Theorem rexdifsn 3800

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 3795	. . . 4 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2	1	anbi1i 458	. . 3 |- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( ( x e. A /\ x =/= B ) /\ ph ) )
3		anass 401	. . 3 |- ( ( ( x e. A /\ x =/= B ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
4	2, 3	bitri 184	. 2 |- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
5	4	rexbii2 2541	1 |- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2200   =/= wne 2400   E.wrex 2509   \ cdif 3194   {csn 3666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-rex 2514  df-v 2801  df-dif 3199  df-sn 3672

This theorem is referenced by: (None)