Theorem rexdifsn 3776

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 3771	. . . 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 2519	1 |- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2178   =/= wne 2378   E.wrex 2487   \ cdif 3171   {csn 3643

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-rex 2492  df-v 2778  df-dif 3176  df-sn 3649

This theorem is referenced by: (None)