Theorem mo2icl 1434

Description: Theorem for inferring "at most one".

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem mo2icl

Step	Hyp	Ref	Expression
1		cleq2 1110	. . . . . 6 |- (y = A -> (x = y <-> x = A))
2	1	imbi2d 464	. . . . 5 |- (y = A -> ((ph -> x = y) <-> (ph -> x = A)))
3	2	bialdv 935	. . . 4 |- (y = A -> (A.x(ph -> x = y) <-> A.x(ph -> x = A)))
4	3	imbi1d 465	. . 3 |- (y = A -> ((A.x(ph -> x = y) -> E*xph) <-> (A.x(ph -> x = A) -> E*xph)))
5		19.8a 712	. . . 4 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
6		ax-17 925	. . . . 5 |- (ph -> A.yph)
7	6	mo2 1026	. . . 4 |- (E*xph <-> E.yA.x(ph -> x = y))
8	5, 7	sylibr 175	. . 3 |- (A.x(ph -> x = y) -> E*xph)
9	4, 8	vtoclg 1383	. 2 |- (A e. V -> (A.x(ph -> x = A) -> E*xph))
10		visset 1350	. . . . . . . 8 |- x e. V
11		eleq1 1149	. . . . . . . 8 |- (x = A -> (x e. V <-> A e. V))
12	10, 11	mpbii 168	. . . . . . 7 |- (x = A -> A e. V)
13	12	syl3 18	. . . . . 6 |- ((ph -> x = A) -> (ph -> A e. V))
14	13	con3d 87	. . . . 5 |- ((ph -> x = A) -> (-. A e. V -> -. ph))
15	14	com12 13	. . . 4 |- (-. A e. V -> ((ph -> x = A) -> -. ph))
16	15	19.20dv 946	. . 3 |- (-. A e. V -> (A.x(ph -> x = A) -> A.x -. ph))
17		alnex 716	. . . 4 |- (A.x -. ph <-> -. E.xph)
18		exmo 1042	. . . . 5 |- (E.xph \/ E*xph)
19	18	ori 200	. . . 4 |- (-. E.xph -> E*xph)
20	17, 19	sylbi 174	. . 3 |- (A.x -. ph -> E*xph)
21	16, 20	syl6 23	. 2 |- (-. A e. V -> (A.x(ph -> x = A) -> E*xph))
22	9, 21	pm2.61i 110	1 |- (A.x(ph -> x = A) -> E*xph)

Syntax hints:  -. wn 1   -> wi 2  A.wal 672  E.wex 678   = weq 797  E*wmo 1008   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  aceq6b 3565

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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