Theorem mopick 1975

Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Assertion

Ref	Expression
mopick	|- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Proof of Theorem mopick

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1416	. . . 4 |- ((ph /\ ps) -> A.y(ph /\ ps))
2		hbs1 1811	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
3		hbs1 1811	. . . . 5 |- ([y / x]ps -> A.x[y / x]ps)
4	2, 3	hban 1436	. . . 4 |- (([y / x]ph /\ [y / x]ps) -> A.x([y / x]ph /\ [y / x]ps))
5		sbequ12 1651	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
6		sbequ12 1651	. . . . 5 |- (x = y -> (ps <-> [y / x]ps))
7	5, 6	anbi12d 442	. . . 4 |- (x = y -> ((ph /\ ps) <-> ([y / x]ph /\ [y / x]ps)))
8	1, 4, 7	cbvexh 1635	. . 3 |- (E.x(ph /\ ps) <-> E.y([y / x]ph /\ [y / x]ps))
9		ax-17 1416	. . . . . . 7 |- (ph -> A.yph)
10	9	mo3h 1950	. . . . . 6 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
11		ax-4 1397	. . . . . . 7 |- (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
12	11	sps 1427	. . . . . 6 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
13	10, 12	sylbi 114	. . . . 5 |- (E*xph -> ((ph /\ [y / x]ph) -> x = y))
14		sbequ2 1649	. . . . . . . . 9 |- (x = y -> ([y / x]ps -> ps))
15	14	imim2i 12	. . . . . . . 8 |- (((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> ([y / x]ps -> ps)))
16	15	expd 245	. . . . . . 7 |- (((ph /\ [y / x]ph) -> x = y) -> (ph -> ([y / x]ph -> ([y / x]ps -> ps))))
17	16	com4t 79	. . . . . 6 |- ([y / x]ph -> ([y / x]ps -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps))))
18	17	imp 115	. . . . 5 |- (([y / x]ph /\ [y / x]ps) -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps)))
19	13, 18	syl5 28	. . . 4 |- (([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
20	19	exlimiv 1486	. . 3 |- (E.y([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
21	8, 20	sylbi 114	. 2 |- (E.x(ph /\ ps) -> (E*xph -> (ph -> ps)))
22	21	impcom 116	1 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240  E.wex 1378  [wsb 1642  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-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by:  eupick  1976  mopick2  1980  moexexdc  1981  euexex  1982  morex  2719  imadif  4922  funimaexglem  4925