Theorem mopick2 1439

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 1096.

Assertion

Ref	Expression
mopick2	|- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))

Proof of Theorem mopick2

Step	Hyp	Ref	Expression
1		pm3.26 319	. . . 4 |- ((ph /\ ps) -> ph)
2	1	19.22i 1042	. . 3 |- (E.x(ph /\ ps) -> E.xph)
3	2	3ad2ant2 803	. 2 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.xph)
4		hbmo1 1408	. . . 4 |- (E*xph -> A.xE*xph)
5		hbe1 1018	. . . 4 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
6		hbe1 1018	. . . 4 |- (E.x(ph /\ ch) -> A.xE.x(ph /\ ch))
7	4, 5, 6	hb3an 1014	. . 3 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> A.x(E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)))
8		mopick 1435	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
9		mopick 1435	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ch)) -> (ph -> ch))
10	8, 9	anim12i 333	. . . . . 6 |- (((E*xph /\ E.x(ph /\ ps)) /\ (E*xph /\ E.x(ph /\ ch))) -> ((ph -> ps) /\ (ph -> ch)))
11		3anass 781	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) <-> (E*xph /\ (E.x(ph /\ ps) /\ E.x(ph /\ ch))))
12		anandi 512	. . . . . . 7 |- ((E*xph /\ (E.x(ph /\ ps) /\ E.x(ph /\ ch))) <-> ((E*xph /\ E.x(ph /\ ps)) /\ (E*xph /\ E.x(ph /\ ch))))
13	11, 12	bitr 173	. . . . . 6 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) <-> ((E*xph /\ E.x(ph /\ ps)) /\ (E*xph /\ E.x(ph /\ ch))))
14		jcab 600	. . . . . 6 |- ((ph -> (ps /\ ch)) <-> ((ph -> ps) /\ (ph -> ch)))
15	10, 13, 14	3imtr4 219	. . . . 5 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (ph -> (ps /\ ch)))
16	15	ancld 298	. . . 4 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (ph -> (ph /\ (ps /\ ch))))
17		3anass 781	. . . 4 |- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
18	16, 17	syl6ibr 213	. . 3 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (ph -> (ph /\ ps /\ ch)))
19	7, 18	19.22d 1064	. 2 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> (E.xph -> E.x(ph /\ ps /\ ch)))
20	3, 19	mpd 26	1 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.x(ph /\ ps /\ ch))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777  E.wex 982  E*wmo 1383

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385

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