Theorem rexdifsn 4323

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 4317	. . . 4 |- ( x e. ( A \ { B } ) <-> ( x e. A /\ x =/= B ) )
2	1	anbi1i 731	. . 3 |- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( ( x e. A /\ x =/= B ) /\ ph ) )
3		anass 681	. . 3 |- ( ( ( x e. A /\ x =/= B ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
4	2, 3	bitri 264	. 2 |- ( ( x e. ( A \ { B } ) /\ ph ) <-> ( x e. A /\ ( x =/= B /\ ph ) ) )
5	4	rexbii2 3039	1 |- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by:  symgfix2  17836  usgr2pth0  26661  wspniunwspnon  26819  dihatexv  36627  lcfl8b  36793