Theorem moexex 2541

Description: "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)

Hypothesis

Ref	Expression
moexex.1	|- F/ y ph

Assertion

Ref	Expression
moexex	|- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )

Proof of Theorem moexex

Step	Hyp	Ref	Expression
1		nfmo1 2481	. . . 4 |- F/ x E* x ph
2		nfa1 2028	. . . . 5 |- F/ x A. x E* y ps
3		nfe1 2027	. . . . . 6 |- F/ x E. x ( ph /\ ps )
4	3	nfmo 2487	. . . . 5 |- F/ x E* y E. x ( ph /\ ps )
5	2, 4	nfim 1825	. . . 4 |- F/ x ( A. x E* y ps -> E* y E. x ( ph /\ ps ) )
6		moexex.1	. . . . . . 7 |- F/ y ph
7	6	nfmo 2487	. . . . . 6 |- F/ y E* x ph
8		mopick 2535	. . . . . . . 8 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
9	8	ex 450	. . . . . . 7 |- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
10	9	com23 86	. . . . . 6 |- ( E* x ph -> ( ph -> ( E. x ( ph /\ ps ) -> ps ) ) )
11	7, 6, 10	alrimd 2084	. . . . 5 |- ( E* x ph -> ( ph -> A. y ( E. x ( ph /\ ps ) -> ps ) ) )
12		moim 2519	. . . . . 6 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( E* y ps -> E* y E. x ( ph /\ ps ) ) )
13	12	spsd 2057	. . . . 5 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
14	11, 13	syl6 35	. . . 4 |- ( E* x ph -> ( ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
15	1, 5, 14	exlimd 2087	. . 3 |- ( E* x ph -> ( E. x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
16	6	nfex 2154	. . . . . . 7 |- F/ y E. x ph
17		exsimpl 1795	. . . . . . 7 |- ( E. x ( ph /\ ps ) -> E. x ph )
18	16, 17	exlimi 2086	. . . . . 6 |- ( E. y E. x ( ph /\ ps ) -> E. x ph )
19		exmo 2495	. . . . . . 7 |- ( E. y E. x ( ph /\ ps ) \/ E* y E. x ( ph /\ ps ) )
20	19	ori 390	. . . . . 6 |- ( -. E. y E. x ( ph /\ ps ) -> E* y E. x ( ph /\ ps ) )
21	18, 20	nsyl4 156	. . . . 5 |- ( -. E* y E. x ( ph /\ ps ) -> E. x ph )
22	21	con1i 144	. . . 4 |- ( -. E. x ph -> E* y E. x ( ph /\ ps ) )
23	22	a1d 25	. . 3 |- ( -. E. x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
24	15, 23	pm2.61d1 171	. 2 |- ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
25	24	imp 445	1 |- ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   E*wmo 2471

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  moexexv  2542  2moswap  2547