Theorem exval 143

Description: Value of the 'there exists' predicate. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypothesis

Ref	Expression
alval.1	|- F:(al -> *)

Assertion

Ref	Expression
exval	|- T. |= [(E.F) = (A.\q:* [(A.\x:al [(Fx:al) ==> q:*]) ==> q:*])]

Distinct variable groups:   x,q,al   q,F,x

Proof of Theorem exval

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wex 139	. . 3 |- E.:((al -> *) -> *)
2		alval.1	. . 3 |- F:(al -> *)
3	1, 2	wc 50	. 2 |- (E.F):*
4		df-ex 131	. . 3 |- T. |= [E. = \p:(al -> *) (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*])]
5	1, 2, 4	ceq1 89	. 2 |- T. |= [(E.F) = (\p:(al -> *) (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*])F)]
6		wal 134	. . . 4 |- A.:((* -> *) -> *)
7		wim 137	. . . . . 6 |- ==> :(* -> (* -> *))
8		wal 134	. . . . . . 7 |- A.:((al -> *) -> *)
9		wv 64	. . . . . . . . . 10 |- p:(al -> *):(al -> *)
10		wv 64	. . . . . . . . . 10 |- x:al:al
11	9, 10	wc 50	. . . . . . . . 9 |- (p:(al -> *)x:al):*
12		wv 64	. . . . . . . . 9 |- q:*:*
13	7, 11, 12	wov 72	. . . . . . . 8 |- [(p:(al -> *)x:al) ==> q:*]:*
14	13	wl 66	. . . . . . 7 |- \x:al [(p:(al -> *)x:al) ==> q:*]:(al -> *)
15	8, 14	wc 50	. . . . . 6 |- (A.\x:al [(p:(al -> *)x:al) ==> q:*]):*
16	7, 15, 12	wov 72	. . . . 5 |- [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:*
17	16	wl 66	. . . 4 |- \q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]:(* -> *)
18	6, 17	wc 50	. . 3 |- (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]):*
19	9, 2	weqi 76	. . . . . . . . . . 11 |- [p:(al -> *) = F]:*
20	19	id 25	. . . . . . . . . 10 |- [p:(al -> *) = F] |= [p:(al -> *) = F]
21	9, 10, 20	ceq1 89	. . . . . . . . 9 |- [p:(al -> *) = F] |= [(p:(al -> *)x:al) = (Fx:al)]
22	7, 11, 12, 21	oveq1 99	. . . . . . . 8 |- [p:(al -> *) = F] |= [[(p:(al -> *)x:al) ==> q:*] = [(Fx:al) ==> q:*]]
23	13, 22	leq 91	. . . . . . 7 |- [p:(al -> *) = F] |= [\x:al [(p:(al -> *)x:al) ==> q:*] = \x:al [(Fx:al) ==> q:*]]
24	8, 14, 23	ceq2 90	. . . . . 6 |- [p:(al -> *) = F] |= [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) = (A.\x:al [(Fx:al) ==> q:*])]
25	7, 15, 12, 24	oveq1 99	. . . . 5 |- [p:(al -> *) = F] |= [[(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*] = [(A.\x:al [(Fx:al) ==> q:*]) ==> q:*]]
26	16, 25	leq 91	. . . 4 |- [p:(al -> *) = F] |= [\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*] = \q:* [(A.\x:al [(Fx:al) ==> q:*]) ==> q:*]]
27	6, 17, 26	ceq2 90	. . 3 |- [p:(al -> *) = F] |= [(A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*]) = (A.\q:* [(A.\x:al [(Fx:al) ==> q:*]) ==> q:*])]
28	18, 2, 27	cl 116	. 2 |- T. |= [(\p:(al -> *) (A.\q:* [(A.\x:al [(p:(al -> *)x:al) ==> q:*]) ==> q:*])F) = (A.\q:* [(A.\x:al [(Fx:al) ==> q:*]) ==> q:*])]
29	3, 5, 28	eqtri 95	1 |- T. |= [(E.F) = (A.\q:* [(A.\x:al [(Fx:al) ==> q:*]) ==> q:*])]

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  E.tex 123

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-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129  df-ex 131

This theorem is referenced by:  exlimdv2  166  ax4e  168  exlimd  183