Theorem prtlem100 32423

Description: Lemma for prter3 32448. (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 654	. . 3 |- ( ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
2		eldifsn 4096	. . . 4 |- ( x e. ( A \ { (/) } ) <-> ( x e. A /\ x =/= (/) ) )
3	2	anbi1i 700	. . 3 |- ( ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) <-> ( ( x e. A /\ x =/= (/) ) /\ ( B e. x /\ ph ) ) )
4		ne0i 3736	. . . . . . 7 |- ( B e. x -> x =/= (/) )
5	4	pm4.71ri 638	. . . . . 6 |- ( B e. x <-> ( x =/= (/) /\ B e. x ) )
6	5	anbi1i 700	. . . . 5 |- ( ( B e. x /\ ph ) <-> ( ( x =/= (/) /\ B e. x ) /\ ph ) )
7		anass 654	. . . . 5 |- ( ( ( x =/= (/) /\ B e. x ) /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
8	6, 7	bitri 253	. . . 4 |- ( ( B e. x /\ ph ) <-> ( x =/= (/) /\ ( B e. x /\ ph ) ) )
9	8	anbi2i 699	. . 3 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. A /\ ( x =/= (/) /\ ( B e. x /\ ph ) ) ) )
10	1, 3, 9	3bitr4ri 282	. 2 |- ( ( x e. A /\ ( B e. x /\ ph ) ) <-> ( x e. ( A \ { (/) } ) /\ ( B e. x /\ ph ) ) )
11	10	rexbii2 2886	1 |- ( E. x e. A ( B e. x /\ ph ) <-> E. x e. ( A \ { (/) } ) ( B e. x /\ ph ) )

Syntax hints:   <-> wb 188   /\ wa 371   e. wcel 1886   =/= wne 2621   E.wrex 2737   \ cdif 3400   (/)c0 3730   {csn 3967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-rex 2742  df-v 3046  df-dif 3406  df-nul 3731  df-sn 3968

This theorem is referenced by: (None)