Theorem euval 144

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

Hypothesis

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

Assertion

Ref	Expression
euval	|- T. |= [(E!F) = (E.\y:al (A.\x:al [(Fx:al) = [x:al = y:al]]))]

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

Proof of Theorem euval

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		weu 141	. . 3 |- E!:((al -> *) -> *)
2		alval.1	. . 3 |- F:(al -> *)
3	1, 2	wc 50	. 2 |- (E!F):*
4		df-eu 133	. . 3 |- T. |= [E! = \p:(al -> *) (E.\y:al (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]]))]
5	1, 2, 4	ceq1 89	. 2 |- T. |= [(E!F) = (\p:(al -> *) (E.\y:al (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]]))F)]
6		wex 139	. . . 4 |- E.:((al -> *) -> *)
7		wal 134	. . . . . 6 |- A.:((al -> *) -> *)
8		wv 64	. . . . . . . . 9 |- p:(al -> *):(al -> *)
9		wv 64	. . . . . . . . 9 |- x:al:al
10	8, 9	wc 50	. . . . . . . 8 |- (p:(al -> *)x:al):*
11		wv 64	. . . . . . . . 9 |- y:al:al
12	9, 11	weqi 76	. . . . . . . 8 |- [x:al = y:al]:*
13	10, 12	weqi 76	. . . . . . 7 |- [(p:(al -> *)x:al) = [x:al = y:al]]:*
14	13	wl 66	. . . . . 6 |- \x:al [(p:(al -> *)x:al) = [x:al = y:al]]:(al -> *)
15	7, 14	wc 50	. . . . 5 |- (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]]):*
16	15	wl 66	. . . 4 |- \y:al (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]]):(al -> *)
17	6, 16	wc 50	. . 3 |- (E.\y:al (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]])):*
18		weq 41	. . . . . . . 8 |- = :(* -> (* -> *))
19	8, 2	weqi 76	. . . . . . . . . 10 |- [p:(al -> *) = F]:*
20	19	id 25	. . . . . . . . 9 |- [p:(al -> *) = F] |= [p:(al -> *) = F]
21	8, 9, 20	ceq1 89	. . . . . . . 8 |- [p:(al -> *) = F] |= [(p:(al -> *)x:al) = (Fx:al)]
22	18, 10, 12, 21	oveq1 99	. . . . . . 7 |- [p:(al -> *) = F] |= [[(p:(al -> *)x:al) = [x:al = y:al]] = [(Fx:al) = [x:al = y:al]]]
23	13, 22	leq 91	. . . . . 6 |- [p:(al -> *) = F] |= [\x:al [(p:(al -> *)x:al) = [x:al = y:al]] = \x:al [(Fx:al) = [x:al = y:al]]]
24	7, 14, 23	ceq2 90	. . . . 5 |- [p:(al -> *) = F] |= [(A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]]) = (A.\x:al [(Fx:al) = [x:al = y:al]])]
25	15, 24	leq 91	. . . 4 |- [p:(al -> *) = F] |= [\y:al (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]]) = \y:al (A.\x:al [(Fx:al) = [x:al = y:al]])]
26	6, 16, 25	ceq2 90	. . 3 |- [p:(al -> *) = F] |= [(E.\y:al (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]])) = (E.\y:al (A.\x:al [(Fx:al) = [x:al = y:al]]))]
27	17, 2, 26	cl 116	. 2 |- T. |= [(\p:(al -> *) (E.\y:al (A.\x:al [(p:(al -> *)x:al) = [x:al = y:al]]))F) = (E.\y:al (A.\x:al [(Fx:al) = [x:al = y:al]]))]
28	3, 5, 27	eqtri 95	1 |- T. |= [(E!F) = (E.\y:al (A.\x:al [(Fx:al) = [x:al = y:al]]))]

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  A.tal 122  E.tex 123  E!teu 125

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  df-eu 133

This theorem is referenced by: (None)