Theorem rexdifsn 3844

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 3840	. . . 4 |- (x e. (A \ {B}) <-> (x e. A /\ x =/= B))
2	1	anbi1i 676	. . 3 |- ((x e. (A \ {B}) /\ ph) <-> ((x e. A /\ x =/= B) /\ ph))
3		anass 630	. . 3 |- (((x e. A /\ x =/= B) /\ ph) <-> (x e. A /\ (x =/= B /\ ph)))
4	2, 3	bitri 240	. 2 |- ((x e. (A \ {B}) /\ ph) <-> (x e. A /\ (x =/= B /\ ph)))
5	4	rexbii2 2644	1 |- (E.x e. (A \ {B})ph <-> E.x e. A (x =/= B /\ ph))

Syntax hints:   <-> wb 176   /\ wa 358   e. wcel 1710   =/= wne 2517  E.wrex 2616   \ cdif 3207  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-sn 3742

This theorem is referenced by: (None)