Theorem mopick 2266

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		nfv 1619	. . . 4 |- F/y(ph /\ ps)
2		nfs1v 2106	. . . . 5 |- F/x[y / x]ph
3		nfs1v 2106	. . . . 5 |- F/x[y / x]ps
4	2, 3	nfan 1824	. . . 4 |- F/x([y / x]ph /\ [y / x]ps)
5		sbequ12 1919	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
6		sbequ12 1919	. . . . 5 |- (x = y -> (ps <-> [y / x]ps))
7	5, 6	anbi12d 691	. . . 4 |- (x = y -> ((ph /\ ps) <-> ([y / x]ph /\ [y / x]ps)))
8	1, 4, 7	cbvex 1985	. . 3 |- (E.x(ph /\ ps) <-> E.y([y / x]ph /\ [y / x]ps))
9		nfv 1619	. . . . . . 7 |- F/yph
10	9	mo3 2235	. . . . . 6 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
11		sp 1747	. . . . . . 7 |- (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
12	11	sps 1754	. . . . . 6 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
13	10, 12	sylbi 187	. . . . 5 |- (E*xph -> ((ph /\ [y / x]ph) -> x = y))
14		sbequ2 1650	. . . . . . . . 9 |- (x = y -> ([y / x]ps -> ps))
15	14	imim2i 13	. . . . . . . 8 |- (((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> ([y / x]ps -> ps)))
16	15	exp3a 425	. . . . . . 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 418	. . . . 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 1634	. . 3 |- (E.y([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
21	8, 20	sylbi 187	. 2 |- (E.x(ph /\ ps) -> (E*xph -> (ph -> ps)))
22	21	impcom 419	1 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  [wsb 1648  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  eupick  2267  mopick2  2271  moexex  2273  morex  3021  imadif  5172