Theorem prtlem100 34144

Description: Lemma for prter3 34167. (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 681	. . 3 |- ( ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
2		eldifsn 4317	. . . 4 |- ( x e. ( A \ { (/) } ) <-> ( x e. A /\ x =/= (/) ) )
3	2	anbi1i 731	. . 3 |- ( ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) <-> ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) )
4		ne0i 3921	. . . . . . 7 |- ( B e. x -> x =/= (/) )
5	4	pm4.71ri 665	. . . . . 6 |- ( B e. x <-> ( x =/= (/) /\ B e. x ) )
6	5	anbi1i 731	. . . . 5 |- ( ( B e. x /\ ph ) <-> ( ( x =/= (/) /\ B e. x ) /\ ph ) )
7		anass 681	. . . . 5 |- ( ( ( x =/= (/) /\ B e. x ) /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
8	6, 7	bitri 264	. . . 4 |- ( ( B e. x /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
9	8	anbi2i 730	. . 3 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
10	1, 3, 9	3bitr4ri 293	. 2 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) )
11	10	rexbii2 3039	1 |- ( E. x e. A ( B e. x /\ ph ) <-> E. x e. ( A \ { (/) } ) ( B e. x /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   (/)c0 3915   {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-nul 3916  df-sn 4178

This theorem is referenced by: (None)