Theorem mopick2 1436

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 1094.

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 1040	. . 3 |- (E.x(ph /\ ps) -> E.xph)
3	2	3ad2ant2 801	. 2 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) -> E.xph)
4		hbmo1 1406	. . . 4 |- (E*xph -> A.xE*xph)
5		hbe1 1016	. . . 4 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
6		hbe1 1016	. . . 4 |- (E.x(ph /\ ch) -> A.xE.x(ph /\ ch))
7	4, 5, 6	hb3an 1012	. . 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 1433	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ps)) -> (ph -> ps))
9		mopick 1433	. . . . . . 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 779	. . . . . . 7 |- ((E*xph /\ E.x(ph /\ ps) /\ E.x(ph /\ ch)) <-> (E*xph /\ (E.x(ph /\ ps) /\ E.x(ph /\ ch))))
12		anandi 510	. . . . . . 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 598	. . . . . 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 779	. . . 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 1062	. 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 775  E.wex 980  E*wmo 1381

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-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383

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