Theorem mo2icl 3016

Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)

Assertion

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

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem mo2icl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2362	. . . . . 6 |- (y = A -> (x = y <-> x = A))
2	1	imbi2d 307	. . . . 5 |- (y = A -> ((ph -> x = y) <-> (ph -> x = A)))
3	2	albidv 1625	. . . 4 |- (y = A -> (A.x(ph -> x = y) <-> A.x(ph -> x = A)))
4	3	imbi1d 308	. . 3 |- (y = A -> ((A.x(ph -> x = y) -> E*xph) <-> (A.x(ph -> x = A) -> E*xph)))
5		19.8a 1756	. . . 4 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
6		nfv 1619	. . . . 5 |- F/yph
7	6	mo2 2233	. . . 4 |- (E*xph <-> E.yA.x(ph -> x = y))
8	5, 7	sylibr 203	. . 3 |- (A.x(ph -> x = y) -> E*xph)
9	4, 8	vtoclg 2915	. 2 |- (A e. _V -> (A.x(ph -> x = A) -> E*xph))
10		vex 2863	. . . . . . 7 |- x e. _V
11		eleq1 2413	. . . . . . 7 |- (x = A -> (x e. _V <-> A e. _V))
12	10, 11	mpbii 202	. . . . . 6 |- (x = A -> A e. _V)
13	12	imim2i 13	. . . . 5 |- ((ph -> x = A) -> (ph -> A e. _V))
14	13	con3rr3 128	. . . 4 |- (-. A e. _V -> ((ph -> x = A) -> -. ph))
15	14	alimdv 1621	. . 3 |- (-. A e. _V -> (A.x(ph -> x = A) -> A.x -. ph))
16		alnex 1543	. . . 4 |- (A.x -. ph <-> -. E.xph)
17		exmo 2249	. . . . 5 |- (E.xph \/ E*xph)
18	17	ori 364	. . . 4 |- (-. E.xph -> E*xph)
19	16, 18	sylbi 187	. . 3 |- (A.x -. ph -> E*xph)
20	15, 19	syl6 29	. 2 |- (-. A e. _V -> (A.x(ph -> x = A) -> E*xph))
21	9, 20	pm2.61i 156	1 |- (A.x(ph -> x = A) -> E*xph)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E*wmo 2205  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by: (None)