Theorem reusv2lem1 4657

Description: Lemma for reusv2 4662. (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 3803	. . 3 |- ( A =/= (/) <-> E. y y e. A )
2		nfra1 2838	. . . . 5 |- F/ y A. y e. A x = B
3	2	nfmo 2302	. . . 4 |- F/ y E* x A. y e. A x = B
4		rsp 2823	. . . . . . 7 |- ( A. y e. A x = B -> ( y e. A -> x = B ) )
5	4	com12 31	. . . . . 6 |- ( y e. A -> ( A. y e. A x = B -> x = B ) )
6	5	alrimiv 1720	. . . . 5 |- ( y e. A -> A. x ( A. y e. A x = B -> x = B ) )
7		moeq 3275	. . . . 5 |- E* x x = B
8		moim 2340	. . . . 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 18	. . . 4 |- ( y e. A -> E* x A. y e. A x = B )
10	3, 9	exlimi 1913	. . 3 |- ( E. y y e. A -> E* x A. y e. A x = B )
11	1, 10	sylbi 195	. 2 |- ( A =/= (/) -> E* x A. y e. A x = B )
12		eu5 2311	. . 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 906	. 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 16	1 |- ( A =/= (/) -> ( E! x A. y e. A x = B <-> E. x A. y e. A x = B ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1393   = wceq 1395   E.wex 1613   e. wcel 1819   E!weu 2283   E*wmo 2284   =/= wne 2652   A.wral 2807   (/)c0 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3474  df-nul 3794

This theorem is referenced by: (None)