Theorem euexex 1985

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 1947	. . 3 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		nfmo1 1912	. . . . . 6 |- F/ x E* x ph
3		nfa1 1434	. . . . . . 7 |- F/ x A. x E* y ps
4		nfe1 1385	. . . . . . . 8 |- F/ x E. x ( ph /\ ps )
5	4	nfmo 1920	. . . . . . 7 |- F/ x E* y E. x ( ph /\ ps )
6	3, 5	nfim 1464	. . . . . 6 |- F/ x ( A. x E* y ps -> E* y E. x ( ph /\ ps ) )
7	2, 6	nfim 1464	. . . . 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 1920	. . . . . . 7 |- F/ y E* x ph
10		mopick 1978	. . . . . . . . 9 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )
11	10	ex 108	. . . . . . . 8 |- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
12	11	com3r 73	. . . . . . 7 |- ( ph -> ( E* x ph -> ( E. x ( ph /\ ps ) -> ps ) ) )
13	8, 9, 12	alrimd 1501	. . . . . 6 |- ( ph -> ( E* x ph -> A. y ( E. x ( ph /\ ps ) -> ps ) ) )
14		moim 1964	. . . . . . 7 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( E* y ps -> E* y E. x ( ph /\ ps ) ) )
15	14	spsd 1431	. . . . . 6 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
16	13, 15	syl6 29	. . . . 5 |- ( ph -> ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
17	7, 16	exlimi 1485	. . . 4 |- ( E. x ph -> ( E* x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) ) )
18	17	imp 115	. . 3 |- ( ( E. x ph /\ E* x ph ) -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
19	1, 18	sylbi 114	. 2 |- ( E! x ph -> ( A. x E* y ps -> E* y E. x ( ph /\ ps ) ) )
20	19	imp 115	1 |- ( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   F/wnf 1349   E.wex 1381   E!weu 1900   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  mosubt  2718  funco  4940