Theorem euexex 2026

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 1988	. . 3 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		nfmo1 1953	. . . . . 6 |- F/ x E* x ph
3		nfa1 1474	. . . . . . 7 |- F/ x A. x E* y ps
4		nfe1 1425	. . . . . . . 8 |- F/ x E. x ( ph /\ ps )
5	4	nfmo 1961	. . . . . . 7 |- F/ x E* y E. x ( ph /\ ps )
6	3, 5	nfim 1504	. . . . . 6 |- F/ x ( A. x E* y ps -> E* y E. x ( ph /\ ps ) )
7	2, 6	nfim 1504	. . . . 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 1961	. . . . . . 7 |- F/ y E* x ph
10		mopick 2019	. . . . . . . . 9 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
11	10	ex 113	. . . . . . . 8 |- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
12	11	com3r 78	. . . . . . 7 |- ( ph -> ( E* x ph -> ( E. x ( ph /\ ps ) -> ps ) ) )
13	8, 9, 12	alrimd 1541	. . . . . 6 |- ( ph -> ( E* x ph -> A. y ( E. x ( ph /\ ps ) -> ps ) ) )
14		moim 2005	. . . . . . 7 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( E* y ps -> E* y E. x ( ph /\ ps ) ) )
15	14	spsd 1471	. . . . . 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 1525	. . . 4 |- ( E. x ph -> ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
18	17	imp 122	. . 3 |- ( ( E. x ph /\ E* x ph ) -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
19	1, 18	sylbi 119	. 2 |- ( E! x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
20	19	imp 122	1 |- ( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   F/wnf 1389   E.wex 1421   E!weu 1941   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  mosubt  2769  funco  4960