Theorem euexex 1982

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

Hypothesis

Ref	Expression
euexex.1	|- F/yph

Assertion

Ref	Expression
euexex	|- ((E!xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))

Proof of Theorem euexex

Step	Hyp	Ref	Expression
1		eu5 1944	. . 3 |- (E!xph <-> (E.xph /\ E*xph))
2		nfmo1 1909	. . . . . 6 |- F/xE*xph
3		nfa1 1431	. . . . . . 7 |- F/xA.xE*yps
4		nfe1 1382	. . . . . . . 8 |- F/xE.x(ph /\ ps)
5	4	nfmo 1917	. . . . . . 7 |- F/xE*yE.x(ph /\ ps)
6	3, 5	nfim 1461	. . . . . 6 |- F/x(A.xE*yps -> E*yE.x(ph /\ ps))
7	2, 6	nfim 1461	. . . . 5 |- F/x(E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
8		euexex.1	. . . . . . 7 |- F/yph
9	8	nfmo 1917	. . . . . . 7 |- F/yE*xph
10		mopick 1975	. . . . . . . . 9 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
11	10	ex 108	. . . . . . . 8 |- (E*xph -> (E.x(ph /\ ps) -> (ph -> ps)))
12	11	com3r 73	. . . . . . 7 |- (ph -> (E*xph -> (E.x(ph /\ ps) -> ps)))
13	8, 9, 12	alrimd 1498	. . . . . 6 |- (ph -> (E*xph -> A.y(E.x(ph /\ ps) -> ps)))
14		moim 1961	. . . . . . 7 |- (A.y(E.x(ph /\ ps) -> ps) -> (E*yps -> E*yE.x(ph /\ ps)))
15	14	spsd 1428	. . . . . 6 |- (A.y(E.x(ph /\ ps) -> ps) -> (A.xE*yps -> E*yE.x(ph /\ ps)))
16	13, 15	syl6 29	. . . . 5 |- (ph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
17	7, 16	exlimi 1482	. . . 4 |- (E.xph -> (E*xph -> (A.xE*yps -> E*yE.x(ph /\ ps))))
18	17	imp 115	. . 3 |- ((E.xph /\ E*xph) -> (A.xE*yps -> E*yE.x(ph /\ ps)))
19	1, 18	sylbi 114	. 2 |- (E!xph -> (A.xE*yps -> E*yE.x(ph /\ ps)))
20	19	imp 115	1 |- ((E!xph /\ A.xE*yps) -> E*yE.x(ph /\ ps))

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   F/wnf 1346  E.wex 1378  E!weu 1897  E*wmo 1898

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by:  mosubt  2712  funco  4883