Theorem mopick2OLD 1838

Description: "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1447.

Assertion

Ref	Expression
mopick2OLD	|- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))

Proof of Theorem mopick2OLD

Step	Hyp	Ref	Expression
1		exsimpl 1461	. . 3 |- (E.x(ph /\ ps) -> E.xph)
2	1	3ad2ant2 898	. 2 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.xph)
3		hbmo1 1802	. . . 4 |- (E*xph -> A.xE*xph)
4		hbe1 1363	. . . 4 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
5		hbe1 1363	. . . 4 |- (E.x(ph /\ ch) -> A.xE.x(ph /\ ch))
6	3, 4, 5	hb3an 1359	. . 3 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> A.x(E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)))
7		mopick 1833	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
8		mopick 1833	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ch)) -> (ph -> ch))
9	7, 8	anim12i 360	. . . . . 6 |- (((E*xph /\ E.x(ph /\ ps)) /\ (E*xph /\ E.x(ph /\ ch))) -> ((ph -> ps) /\ (ph -> ch)))
10		3anass 862	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) <-> (E*xph /\ (E.x(ph /\ ps) /\ E.x(ph /\ ch))))
11		anandi 568	. . . . . . 7 |- ((E*xph /\ (E.x(ph /\ ps) /\ E.x(ph /\ ch))) <-> ((E*xph /\ E.x(ph /\ ps)) /\ (E*xph /\ E.x(ph /\ ch))))
12	10, 11	bitri 190	. . . . . 6 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) <-> ((E*xph /\ E.x(ph /\ ps)) /\ (E*xph /\ E.x(ph /\ ch))))
13		jcab 659	. . . . . 6 |- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))
14	9, 12, 13	3imtr4i 236	. . . . 5 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (ph -> (ps /\ ch)))
15	14	ancld 322	. . . 4 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (ph -> (ph /\ (ps /\ ch))))
16		3anass 862	. . . 4 |- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
17	15, 16	syl6ibr 230	. . 3 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (ph -> (ph /\ ps /\ ch)))
18	6, 17	eximd 1410	. 2 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (E.xph -> E.x(ph /\ ps /\ ch)))
19	2, 18	mpd 29	1 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  E.wex 1326  E*wmo 1772

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-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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