Theorem mo2icl 1914

Description: Theorem for inferring "at most one."

Assertion

Ref	Expression
mo2icl	|- (A.x(ph -> x = A) -> E*xph)

Distinct variable group:   x,A

Proof of Theorem mo2icl

Step	Hyp	Ref	Expression
1		eqeq2 1476	. . . . . 6 |- (y = A -> (x = y <-> x = A))
2	1	imbi2d 610	. . . . 5 |- (y = A -> ((ph -> x = y) <-> (ph -> x = A)))
3	2	albidv 1273	. . . 4 |- (y = A -> (A.x(ph -> x = y) <-> A.x(ph -> x = A)))
4	3	imbi1d 611	. . 3 |- (y = A -> ((A.x(ph -> x = y) -> E*xph) <-> (A.x(ph -> x = A) -> E*xph)))
5		19.8a 1025	. . . 4 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
6		ax-17 968	. . . . 5 |- (ph -> A.yph)
7	6	mo2 1393	. . . 4 |- (E*xph <-> E.yA.x(ph -> x = y))
8	5, 7	sylibr 200	. . 3 |- (A.x(ph -> x = y) -> E*xph)
9	4, 8	vtoclg 1838	. 2 |- (A e. V -> (A.x(ph -> x = A) -> E*xph))
10		visset 1804	. . . . . . . 8 |- x e. V
11		eleq1 1526	. . . . . . . 8 |- (x = A -> (x e. V <-> A e. V))
12	10, 11	mpbii 193	. . . . . . 7 |- (x = A -> A e. V)
13	12	imim2i 17	. . . . . 6 |- ((ph -> x = A) -> (ph -> A e. V))
14	13	con3d 95	. . . . 5 |- ((ph -> x = A) -> (-. A e. V -> -. ph))
15	14	com12 11	. . . 4 |- (-. A e. V -> ((ph -> x = A) -> -. ph))
16	15	19.20dv 1284	. . 3 |- (-. A e. V -> (A.x(ph -> x = A) -> A.x -. ph))
17		alnex 1029	. . . 4 |- (A.x -. ph <-> -. E.xph)
18		exmo 1409	. . . . 5 |- (E.xph \/ E*xph)
19	18	ori 230	. . . 4 |- (-. E.xph -> E*xph)
20	17, 19	sylbi 199	. . 3 |- (A.x -. ph -> E*xph)
21	16, 20	syl6 22	. 2 |- (-. A e. V -> (A.x(ph -> x = A) -> E*xph))
22	9, 21	pm2.61i 126	1 |- (A.x(ph -> x = A) -> E*xph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  E*wmo 1374  Vcvv 1802

This theorem is referenced by:  aceq6b 4714

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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