Theorem mopick 1433

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-17 971	. . . 4 |- ((ph /\ ps) -> A.y(ph /\ ps))
2		hbs1 1332	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
3		hbs1 1332	. . . . 5 |- ([y / x]ps -> A.x[y / x]ps)
4	2, 3	hban 1009	. . . 4 |- (([y / x]ph /\ [y / x]ps) -> A.x([y / x]ph /\ [y / x]ps))
5		sbequ12 1181	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
6		sbequ12 1181	. . . . 5 |- (x = y -> (ps <-> [y / x]ps))
7	5, 6	anbi12d 628	. . . 4 |- (x = y -> ((ph /\ ps) <-> ([y / x]ph /\ [y / x]ps)))
8	1, 4, 7	cbvex 1166	. . 3 |- (E.x(ph /\ ps) <-> E.y([y / x]ph /\ [y / x]ps))
9		sbequ2 1179	. . . . . . . . 9 |- (x = y -> ([y / x]ps -> ps))
10	9	imim2i 17	. . . . . . . 8 |- (((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> ([y / x]ps -> ps)))
11	10	exp3a 375	. . . . . . 7 |- (((ph /\ [y / x]ph) -> x = y) -> (ph -> ([y / x]ph -> ([y / x]ps -> ps))))
12	11	com4t 40	. . . . . 6 |- ([y / x]ph -> ([y / x]ps -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps))))
13	12	imp 350	. . . . 5 |- (([y / x]ph /\ [y / x]ps) -> (((ph /\ [y / x]ph) -> x = y) -> (ph -> ps)))
14		ax-17 971	. . . . . . 7 |- (ph -> A.yph)
15	14	mo3 1401	. . . . . 6 |- (E*xph <-> A.xA.y((ph /\ [y / x]ph) -> x = y))
16		ax-4 973	. . . . . . 7 |- (A.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
17	16	a4s 984	. . . . . 6 |- (A.xA.y((ph /\ [y / x]ph) -> x = y) -> ((ph /\ [y / x]ph) -> x = y))
18	15, 17	sylbi 199	. . . . 5 |- (E*xph -> ((ph /\ [y / x]ph) -> x = y))
19	13, 18	syl5 21	. . . 4 |- (([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
20	19	19.23aiv 1295	. . 3 |- (E.y([y / x]ph /\ [y / x]ps) -> (E*xph -> (ph -> ps)))
21	8, 20	sylbi 199	. 2 |- (E.x(ph /\ ps) -> (E*xph -> (ph -> ps)))
22	21	impcom 351	1 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954  E.wex 980  E*wmo 1381

This theorem is referenced by:  eupick 1434  mopick2 1436  moexex 1438  imadif 3574

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383

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