Theorem euexex 2165

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 2127	. . 3 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		nfmo1 2091	. . . . . 6 |- F/ x E* x ph
3		nfa1 1590	. . . . . . 7 |- F/ x A. x E* y ps
4		nfe1 1545	. . . . . . . 8 |- F/ x E. x ( ph /\ ps )
5	4	nfmo 2099	. . . . . . 7 |- F/ x E* y E. x ( ph /\ ps )
6	3, 5	nfim 1621	. . . . . 6 |- F/ x ( A. x E* y ps -> E* y E. x ( ph /\ ps ) )
7	2, 6	nfim 1621	. . . . 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 2099	. . . . . . 7 |- F/ y E* x ph
10		mopick 2158	. . . . . . . . 9 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
11	10	ex 115	. . . . . . . 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 1659	. . . . . 6 |- ( ph -> ( E* x ph -> A. y ( E. x ( ph /\ ps ) -> ps ) ) )
14		moim 2144	. . . . . . 7 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( E* y ps -> E* y E. x ( ph /\ ps ) ) )
15	14	spsd 1587	. . . . . 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 1643	. . . 4 |- ( E. x ph -> ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
18	17	imp 124	. . 3 |- ( ( E. x ph /\ E* x ph ) -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
19	1, 18	sylbi 121	. 2 |- ( E! x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
20	19	imp 124	1 |- ( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   F/wnf 1509   E.wex 1541   E!weu 2079   E*wmo 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083

This theorem is referenced by:  mosubt  2984  funco  5373