Theorem r19.2zb 4061

Description: A response to the notion that the condition A =/= (/) can be removed in r19.2z 4060. Interestingly enough, ph does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)

Assertion

Ref	Expression
r19.2zb	|- ( A =/= (/) <-> ( A. x e. A ph -> E. x e. A ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem r19.2zb

Step	Hyp	Ref	Expression
1		r19.2z 4060	. . 3 |- ( ( A =/= (/) /\ A. x e. A ph ) -> E. x e. A ph )
2	1	ex 450	. 2 |- ( A =/= (/) -> ( A. x e. A ph -> E. x e. A ph ) )
3		noel 3919	. . . . . . 7 |- -. x e. (/)
4	3	pm2.21i 116	. . . . . 6 |- ( x e. (/) -> ph )
5	4	rgen 2922	. . . . 5 |- A. x e. (/) ph
6		raleq 3138	. . . . 5 |- ( A = (/) -> ( A. x e. A ph <-> A. x e. (/) ph ) )
7	5, 6	mpbiri 248	. . . 4 |- ( A = (/) -> A. x e. A ph )
8	7	necon3bi 2820	. . 3 |- ( -. A. x e. A ph -> A =/= (/) )
9		exsimpl 1795	. . . 4 |- ( E. x ( x e. A /\ ph ) -> E. x x e. A )
10		df-rex 2918	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
11		n0 3931	. . . 4 |- ( A =/= (/) <-> E. x x e. A )
12	9, 10, 11	3imtr4i 281	. . 3 |- ( E. x e. A ph -> A =/= (/) )
13	8, 12	ja 173	. 2 |- ( ( A. x e. A ph -> E. x e. A ph ) -> A =/= (/) )
14	2, 13	impbii 199	1 |- ( A =/= (/) <-> ( A. x e. A ph -> E. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915

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-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  iinpreima  6345  utopbas  22039  clsk3nimkb  38338  radcnvrat  38513