Theorem euexex 2111

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 2073	. . 3 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		nfmo1 2038	. . . . . 6 |- F/ x E* x ph
3		nfa1 1541	. . . . . . 7 |- F/ x A. x E* y ps
4		nfe1 1496	. . . . . . . 8 |- F/ x E. x ( ph /\ ps )
5	4	nfmo 2046	. . . . . . 7 |- F/ x E* y E. x ( ph /\ ps )
6	3, 5	nfim 1572	. . . . . 6 |- F/ x ( A. x E* y ps -> E* y E. x ( ph /\ ps ) )
7	2, 6	nfim 1572	. . . . 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 2046	. . . . . . 7 |- F/ y E* x ph
10		mopick 2104	. . . . . . . . 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 1610	. . . . . 6 |- ( ph -> ( E* x ph -> A. y ( E. x ( ph /\ ps ) -> ps ) ) )
14		moim 2090	. . . . . . 7 |- ( A. y ( E. x ( ph /\ ps ) -> ps ) -> ( E* y ps -> E* y E. x ( ph /\ ps ) ) )
15	14	spsd 1538	. . . . . 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 1594	. . . 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 1351   F/wnf 1460   E.wex 1492   E!weu 2026   E*wmo 2027

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030

This theorem is referenced by:  mosubt  2916  funco  5258