Theorem ac 184

Description: Defining property of the indefinite descriptor: it selects an element from any type. This is equivalent to global choice in ZF.

Hypotheses

Ref	Expression
ac.1	|- F:(al -> *)
ac.2	|- A:al

Assertion

Ref	Expression
ac	|- (FA) |= (F(@F))

Proof of Theorem ac

Dummy variables x p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ac.1	. . 3 |- F:(al -> *)
2		wat 180	. . . 4 |- @:((al -> *) -> al)
3	2, 1	wc 45	. . 3 |- (@F):al
4	1, 3	wc 45	. 2 |- (F(@F)):*
5		ac.2	. . . 4 |- A:al
6	1, 5	wc 45	. . 3 |- (FA):*
7	6	id 25	. 2 |- (FA) |= (FA)
8	7	ax-cb1 29	. . 3 |- (FA):*
9		ax-ac 183	. . . . 5 |- T. |= (A.\p:(al -> *) (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]))
10		wal 124	. . . . . . 7 |- A.:((al -> *) -> *)
11		wim 127	. . . . . . . . 9 |- ==> :(* -> (* -> *))
12		wv 58	. . . . . . . . . 10 |- p:(al -> *):(al -> *)
13		wv 58	. . . . . . . . . 10 |- x:al:al
14	12, 13	wc 45	. . . . . . . . 9 |- (p:(al -> *)x:al):*
15	2, 12	wc 45	. . . . . . . . . 10 |- (@p:(al -> *)):al
16	12, 15	wc 45	. . . . . . . . 9 |- (p:(al -> *)(@p:(al -> *))):*
17	11, 14, 16	wov 64	. . . . . . . 8 |- [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]:*
18	17	wl 59	. . . . . . 7 |- \x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]:(al -> *)
19	10, 18	wc 45	. . . . . 6 |- (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]):*
20	12, 1	weqi 68	. . . . . . . . . . 11 |- [p:(al -> *) = F]:*
21	20	id 25	. . . . . . . . . 10 |- [p:(al -> *) = F] |= [p:(al -> *) = F]
22	12, 13, 21	ceq1 79	. . . . . . . . 9 |- [p:(al -> *) = F] |= [(p:(al -> *)x:al) = (Fx:al)]
23	2, 12, 21	ceq2 80	. . . . . . . . . 10 |- [p:(al -> *) = F] |= [(@p:(al -> *)) = (@F)]
24	12, 15, 21, 23	ceq12 78	. . . . . . . . 9 |- [p:(al -> *) = F] |= [(p:(al -> *)(@p:(al -> *))) = (F(@F))]
25	11, 14, 16, 22, 24	oveq12 90	. . . . . . . 8 |- [p:(al -> *) = F] |= [[(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))] = [(Fx:al) ==> (F(@F))]]
26	17, 25	leq 81	. . . . . . 7 |- [p:(al -> *) = F] |= [\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))] = \x:al [(Fx:al) ==> (F(@F))]]
27	10, 18, 26	ceq2 80	. . . . . 6 |- [p:(al -> *) = F] |= [(A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]) = (A.\x:al [(Fx:al) ==> (F(@F))])]
28	19, 1, 27	cla4v 142	. . . . 5 |- (A.\p:(al -> *) (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))])) |= (A.\x:al [(Fx:al) ==> (F(@F))])
29	9, 28	syl 16	. . . 4 |- T. |= (A.\x:al [(Fx:al) ==> (F(@F))])
30	17, 25	eqtypi 69	. . . . 5 |- [(Fx:al) ==> (F(@F))]:*
31	1, 13	wc 45	. . . . . 6 |- (Fx:al):*
32	13, 5	weqi 68	. . . . . . . 8 |- [x:al = A]:*
33	32	id 25	. . . . . . 7 |- [x:al = A] |= [x:al = A]
34	1, 13, 33	ceq2 80	. . . . . 6 |- [x:al = A] |= [(Fx:al) = (FA)]
35	11, 31, 4, 34	oveq1 89	. . . . 5 |- [x:al = A] |= [[(Fx:al) ==> (F(@F))] = [(FA) ==> (F(@F))]]
36	30, 5, 35	cla4v 142	. . . 4 |- (A.\x:al [(Fx:al) ==> (F(@F))]) |= [(FA) ==> (F(@F))]
37	29, 36	syl 16	. . 3 |- T. |= [(FA) ==> (F(@F))]
38	8, 37	a1i 28	. 2 |- (FA) |= [(FA) ==> (F(@F))]
39	4, 7, 38	mpd 146	1 |- (FA) |= (F(@F))

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 111  A.tal 112  @tat 179

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-ac 183

This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119

This theorem is referenced by:  dfex2  185  exmid  186