Theorem reusv2lem1 4604

Description: Lemma for reusv2 4609. (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 3741	. . 3 |- ( A =/= (/) <-> E. y y e. A )
2		nfra1 2769	. . . . 5 |- F/ y A. y e. A x = B
3	2	nfmo 2316	. . . 4 |- F/ y E* x A. y e. A x = B
4		rsp 2754	. . . . . . 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 1773	. . . . 5 |- ( y e. A -> A. x ( A. y e. A x = B -> x = B ) )
7		moeq 3214	. . . . 5 |- E* x x = B
8		moim 2348	. . . . 5 |- ( A. x ( A. y e. A x = B -> x = B ) -> ( E* x x = B -> E* x A. y e. A x = B ) )
9	6, 7, 8	mpisyl 21	. . . 4 |- ( y e. A -> E* x A. y e. A x = B )
10	3, 9	exlimi 1995	. . 3 |- ( E. y y e. A -> E* x A. y e. A x = B )
11	1, 10	sylbi 199	. 2 |- ( A =/= (/) -> E* x A. y e. A x = B )
12		eu5 2325	. . 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 ) )
13	12	rbaib 917	. 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 ) )
14	11, 13	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 188   A.wal 1442   = wceq 1444   E.wex 1663   e. wcel 1887   E!weu 2299   E*wmo 2300   =/= wne 2622   A.wral 2737   (/)c0 3731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-v 3047  df-dif 3407  df-nul 3732

This theorem is referenced by: (None)