Theorem prtlem100 32493

Description: Lemma for prter3 32518. (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 661	. . 3 |- ( ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
2		eldifsn 4088	. . . 4 |- ( x e. ( A \ { (/) } ) <-> ( x e. A /\ x =/= (/) ) )
3	2	anbi1i 709	. . 3 |- ( ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) <-> ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) )
4		ne0i 3728	. . . . . . 7 |- ( B e. x -> x =/= (/) )
5	4	pm4.71ri 645	. . . . . 6 |- ( B e. x <-> ( x =/= (/) /\ B e. x ) )
6	5	anbi1i 709	. . . . 5 |- ( ( B e. x /\ ph ) <-> ( ( x =/= (/) /\ B e. x ) /\ ph ) )
7		anass 661	. . . . 5 |- ( ( ( x =/= (/) /\ B e. x ) /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
8	6, 7	bitri 257	. . . 4 |- ( ( B e. x /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
9	8	anbi2i 708	. . 3 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
10	1, 3, 9	3bitr4ri 286	. 2 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) )
11	10	rexbii2 2879	1 |- ( E. x e. A ( B e. x /\ ph ) <-> E. x e. ( A \ { (/) } ) ( B e. x /\ ph ) )

Syntax hints:   <-> wb 189   /\ wa 376   e. wcel 1904   =/= wne 2641   E.wrex 2757   \ cdif 3387   (/)c0 3722   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rex 2762  df-v 3033  df-dif 3393  df-nul 3723  df-sn 3960

This theorem is referenced by: (None)