Theorem mo2icl 1923

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 1484	. . . . . 6 |- (y = A -> (x = y <-> x = A))
2	1	imbi2d 612	. . . . 5 |- (y = A -> ((ph -> x = y) <-> (ph -> x = A)))
3	2	albidv 1278	. . . 4 |- (y = A -> (A.x(ph -> x = y) <-> A.x(ph -> x = A)))
4	3	imbi1d 613	. . 3 |- (y = A -> ((A.x(ph -> x = y) -> E*xph) <-> (A.x(ph -> x = A) -> E*xph)))
5		19.8a 1029	. . . 4 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
6		ax-17 971	. . . . 5 |- (ph -> A.yph)
7	6	mo2 1400	. . . 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 1847	. 2 |- (A e. V -> (A.x(ph -> x = A) -> E*xph))
10		visset 1813	. . . . . . . 8 |- x e. V
11		eleq1 1534	. . . . . . . 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 1289	. . 3 |- (-. A e. V -> (A.x(ph -> x = A) -> A.x -. ph))
17		alnex 1033	. . . 4 |- (A.x -. ph <-> -. E.xph)
18		exmo 1416	. . . . 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 954   = wceq 956   e. wcel 958  E.wex 980  E*wmo 1381  Vcvv 1811

This theorem is referenced by:  aceq6b 4742

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  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

Copyright terms: Public domain