Theorem mopick 1833

Description: "At most one" picks a variable value, eliminating an existential quantifier.

Assertion

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

Proof of Theorem mopick

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . 4 |- ((ph /\ ps) -> A.y(ph /\ ps))
2		hbs1 1722	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
3		hbs1 1722	. . . . 5 |- ([y / x]ps -> A.x[y / x]ps)
4	2, 3	hban 1356	. . . 4 |- (([y / x]ph /\ [y / x]ps) -> A.x([y / x]ph /\ [y / x]ps))
5		sbequ12 1545	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
6		sbequ12 1545	. . . . 5 |- (x = y -> (ps <-> [y / x]ps))
7	5, 6	anbi12d 690	. . . 4 |- (x = y -> ((ph /\ ps) <-> ([y / x]ph /\ [y / x]ps)))
8	1, 4, 7	cbvex 1529	. . 3 |- (E.x(ph /\ ps) <-> E.y([y / x]ph /\ [y / x]ps))
9		sbequ2 1543	. . . . . . . . 9 |- (x = y -> ([y / x]ps -> ps))
10	9	imim2i 11	. . . . . . . 8 |- (((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> ([y / x]ps -> ps)))
11	10	exp3a 405	. . . . . . 7 |- (((ph /\ [y / x]ph) -> x = y) -> (ph -> ([y / x]ph -> ([y / x]ps -> ps))))
12	11	com4t 44	. . . . . 6 |- ([y / x]ph -> ([y / x]ps -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps))))
13	12	imp 377	. . . . 5 |- (([y / x]ph /\ [y / x]ps) -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps)))
14		ax-17 1317	. . . . . . 7 |- (ph -> A.yph)
15	14	mo3 1797	. . . . . 6 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
16		ax-4 1319	. . . . . . 7 |- (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
17	16	a4s 1330	. . . . . 6 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
18	15, 17	sylbi 216	. . . . 5 |- (E*xph -> ((ph /\ [y / x]ph) -> x = y))
19	13, 18	syl5 20	. . . 4 |- (([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
20	19	19.23aiv 1674	. . 3 |- (E.y([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
21	8, 20	sylbi 216	. 2 |- (E.x(ph /\ ps) -> (E*xph -> (ph -> ps)))
22	21	impcom 378	1 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296  E.wex 1326  E*wmo 1772

This theorem is referenced by:  eupick 1834  mopick2 1837  mopick2OLD 1838  moexex 1841  imadif 4493  morex 15662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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