Theorem ac 197

Description: Defining property of the indefinite descriptor: it selects an element from any type. This is equivalent to global choice in ZF. (Contributed by Mario Carneiro, 10-Oct-2014.)

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 193	. . . 4 |- @:((al -> *) -> al)
3	2, 1	wc 50	. . 3 |- (@F):al
4	1, 3	wc 50	. 2 |- (F(@F)):*
5		ac.2	. . . 4 |- A:al
6	1, 5	wc 50	. . 3 |- (FA):*
7	6	id 25	. 2 |- (FA) |= (FA)
8	7	ax-cb1 29	. . 3 |- (FA):*
9		ax-ac 196	. . . . 5 |- T. |= (A.\p:(al -> *) (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]))
10		wal 134	. . . . . . 7 |- A.:((al -> *) -> *)
11		wim 137	. . . . . . . . 9 |- ==> :(* -> (* -> *))
12		wv 64	. . . . . . . . . 10 |- p:(al -> *):(al -> *)
13		wv 64	. . . . . . . . . 10 |- x:al:al
14	12, 13	wc 50	. . . . . . . . 9 |- (p:(al -> *)x:al):*
15	2, 12	wc 50	. . . . . . . . . 10 |- (@p:(al -> *)):al
16	12, 15	wc 50	. . . . . . . . 9 |- (p:(al -> *)(@p:(al -> *))):*
17	11, 14, 16	wov 72	. . . . . . . 8 |- [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]:*
18	17	wl 66	. . . . . . 7 |- \x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]:(al -> *)
19	10, 18	wc 50	. . . . . 6 |- (A.\x:al [(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))]):*
20	12, 1	weqi 76	. . . . . . . . . . 11 |- [p:(al -> *) = F]:*
21	20	id 25	. . . . . . . . . 10 |- [p:(al -> *) = F] |= [p:(al -> *) = F]
22	12, 13, 21	ceq1 89	. . . . . . . . 9 |- [p:(al -> *) = F] |= [(p:(al -> *)x:al) = (Fx:al)]
23	2, 12, 21	ceq2 90	. . . . . . . . . 10 |- [p:(al -> *) = F] |= [(@p:(al -> *)) = (@F)]
24	12, 15, 21, 23	ceq12 88	. . . . . . . . 9 |- [p:(al -> *) = F] |= [(p:(al -> *)(@p:(al -> *))) = (F(@F))]
25	11, 14, 16, 22, 24	oveq12 100	. . . . . . . 8 |- [p:(al -> *) = F] |= [[(p:(al -> *)x:al) ==> (p:(al -> *)(@p:(al -> *)))] = [(Fx:al) ==> (F(@F))]]
26	17, 25	leq 91	. . . . . . 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 90	. . . . . 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 152	. . . . 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 78	. . . . 5 |- [(Fx:al) ==> (F(@F))]:*
31	1, 13	wc 50	. . . . . 6 |- (Fx:al):*
32	13, 5	weqi 76	. . . . . . . 8 |- [x:al = A]:*
33	32	id 25	. . . . . . 7 |- [x:al = A] |= [x:al = A]
34	1, 13, 33	ceq2 90	. . . . . 6 |- [x:al = A] |= [(Fx:al) = (FA)]
35	11, 31, 4, 34	oveq1 99	. . . . 5 |- [x:al = A] |= [[(Fx:al) ==> (F(@F))] = [(FA) ==> (F(@F))]]
36	30, 5, 35	cla4v 152	. . . 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 156	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 121  A.tal 122  @tat 191

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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-wat 192  ax-ac 196

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129

This theorem is referenced by:  dfex2  198  exmid  199