Theorem reusv2lem1 4868

Description: Lemma for reusv2 4874. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reusv2lem1	|- ( A =/= (/) -> ( E! x A. y e. A x = B <-> E. x A. y e. A x = B ) )

Distinct variable groups:   x, y, A   x, B

Allowed substitution hint:   B( y)

Proof of Theorem reusv2lem1

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( A =/= (/) <-> E. y y e. A )
2		nfra1 2941	. . . . 5 |- F/ y A. y e. A x = B
3	2	nfmo 2487	. . . 4 |- F/ y E* x A. y e. A x = B
4		rsp 2929	. . . . . . 7 |- ( A. y e. A x = B -> ( y e. A -> x = B ) )
5	4	com12 32	. . . . . 6 |- ( y e. A -> ( A. y e. A x = B -> x = B ) )
6	5	alrimiv 1855	. . . . 5 |- ( y e. A -> A. x ( A. y e. A x = B -> x = B ) )
7		mo2icl 3385	. . . . 5 |- ( A. x ( A. y e. A x = B -> x = B ) -> E* x A. y e. A x = B )
8	6, 7	syl 17	. . . 4 |- ( y e. A -> E* x A. y e. A x = B )
9	3, 8	exlimi 2086	. . 3 |- ( E. y y e. A -> E* x A. y e. A x = B )
10	1, 9	sylbi 207	. 2 |- ( A =/= (/) -> E* x A. y e. A x = B )
11		eu5 2496	. . 3 |- ( E! x A. y e. A x = B <-> ( E. x A. y e. A x = B /\ E* x A. y e. A x = B ) )
12	11	rbaib 947	. 2 |- ( E* x A. y e. A x = B -> ( E! x A. y e. A x = B <-> E. x A. y e. A x = B ) )
13	10, 12	syl 17	1 |- ( A =/= (/) -> ( E! x A. y e. A x = B <-> E. x A. y e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   =/= wne 2794   A.wral 2912   (/)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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)