Theorem euexex 2085

Description: Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)

Hypothesis

Ref	Expression
euexex.1	|- F/ y ph

Assertion

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

Proof of Theorem euexex

Step	Hyp	Ref	Expression
1		eu5 2047	. . 3 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		nfmo1 2012	. . . . . 6 |- F/ x E* x ph
3		nfa1 1522	. . . . . . 7 |- F/ x A. x E* y ps
4		nfe1 1473	. . . . . . . 8 |- F/ x E. x ( ph /\ ps )
5	4	nfmo 2020	. . . . . . 7 |- F/ x E* y E. x ( ph /\ ps )
6	3, 5	nfim 1552	. . . . . 6 |- F/ x ( A. x E* y ps -> E* y E. x ( ph /\ ps ) )
7	2, 6	nfim 1552	. . . . 5 |- F/ x ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
8		euexex.1	. . . . . . 7 |- F/ y ph
9	8	nfmo 2020	. . . . . . 7 |- F/ y E* x ph
10		mopick 2078	. . . . . . . . 9 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
11	10	ex 114	. . . . . . . 8 |- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
12	11	com3r 79	. . . . . . 7 |- ( ph -> ( E* x ph -> ( E. x ( ph /\ ps ) -> ps ) ) )
13	8, 9, 12	alrimd 1590	. . . . . 6 |- ( ph -> ( E* x ph -> A. y ( E. x ( ph /\ ps ) -> ps ) ) )
14		moim 2064	. . . . . . 7 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( E* y ps -> E* y E. x ( ph /\ ps ) ) )
15	14	spsd 1519	. . . . . 6 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
16	13, 15	syl6 33	. . . . 5 |- ( ph -> ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
17	7, 16	exlimi 1574	. . . 4 |- ( E. x ph -> ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
18	17	imp 123	. . 3 |- ( ( E. x ph /\ E* x ph ) -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
19	1, 18	sylbi 120	. 2 |- ( E! x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
20	19	imp 123	1 |- ( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   F/wnf 1437   E.wex 1469   E!weu 2000   E*wmo 2001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by:  mosubt  2865  funco  5171