Theorem prtlem100 30762

Description: Lemma for prter3 30789. (Contributed by Rodolfo Medina, 19-Oct-2010.)

Assertion

Ref	Expression
prtlem100	|- ( E. x e. A ( B e. x /\ ph ) <-> E. x e. ( A \ { (/) } ) ( B e. x /\ ph ) )

Proof of Theorem prtlem100

Step	Hyp	Ref	Expression
1		anass 647	. . 3 |- ( ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
2		eldifsn 4069	. . . 4 |- ( x e. ( A \ { (/) } ) <-> ( x e. A /\ x =/= (/) ) )
3	2	anbi1i 693	. . 3 |- ( ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) <-> ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) )
4		ne0i 3717	. . . . . . 7 |- ( B e. x -> x =/= (/) )
5	4	pm4.71ri 631	. . . . . 6 |- ( B e. x <-> ( x =/= (/) /\ B e. x ) )
6	5	anbi1i 693	. . . . 5 |- ( ( B e. x /\ ph ) <-> ( ( x =/= (/) /\ B e. x ) /\ ph ) )
7		anass 647	. . . . 5 |- ( ( ( x =/= (/) /\ B e. x ) /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
8	6, 7	bitri 249	. . . 4 |- ( ( B e. x /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
9	8	anbi2i 692	. . 3 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
10	1, 3, 9	3bitr4ri 278	. 2 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) )
11	10	rexbii2 2882	1 |- ( E. x e. A ( B e. x /\ ph ) <-> E. x e. ( A \ { (/) } ) ( B e. x /\ ph ) )

Syntax hints:   <-> wb 184   /\ wa 367   e. wcel 1826   =/= wne 2577   E.wrex 2733   \ cdif 3386   (/)c0 3711   {csn 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rex 2738  df-v 3036  df-dif 3392  df-nul 3712  df-sn 3945

This theorem is referenced by: (None)