Theorem alnex 174

Description: Theorem 19.7 of [Margaris] p. 89.

Hypothesis

Ref	Expression
alnex1.1	|- A:*

Assertion

Ref	Expression
alnex	|- T. |= [(A.\x:al (~ A)) = (~ (E.\x:al A))]

Distinct variable group:   al,x

Proof of Theorem alnex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alnex1.1	. . . . . 6 |- A:*
2		wfal 125	. . . . . 6 |- F.:*
3		wnot 128	. . . . . . . . 9 |- ~ :(* -> *)
4	3, 1	wc 45	. . . . . . . 8 |- (~ A):*
5	4	ax4 140	. . . . . . 7 |- (A.\x:al (~ A)) |= (~ A)
6	5	ax-cb1 29	. . . . . . . 8 |- (A.\x:al (~ A)):*
7	1	notval 135	. . . . . . . 8 |- T. |= [(~ A) = [A ==> F.]]
8	6, 7	a1i 28	. . . . . . 7 |- (A.\x:al (~ A)) |= [(~ A) = [A ==> F.]]
9	5, 8	mpbi 72	. . . . . 6 |- (A.\x:al (~ A)) |= [A ==> F.]
10	1, 2, 9	imp 147	. . . . 5 |- ((A.\x:al (~ A)), A) |= F.
11		wal 124	. . . . . 6 |- A.:((al -> *) -> *)
12	4	wl 59	. . . . . 6 |- \x:al (~ A):(al -> *)
13		wv 58	. . . . . 6 |- y:al:al
14	11, 13	ax-17 95	. . . . . 6 |- T. |= [(\x:al A.y:al) = A.]
15	4, 13	ax-hbl1 93	. . . . . 6 |- T. |= [(\x:al \x:al (~ A)y:al) = \x:al (~ A)]
16	11, 12, 13, 14, 15	hbc 100	. . . . 5 |- T. |= [(\x:al (A.\x:al (~ A))y:al) = (A.\x:al (~ A))]
17	2, 13	ax-17 95	. . . . 5 |- T. |= [(\x:al F.y:al) = F.]
18	10, 16, 17	exlimd 171	. . . 4 |- ((A.\x:al (~ A)), (E.\x:al A)) |= F.
19	18	ex 148	. . 3 |- (A.\x:al (~ A)) |= [(E.\x:al A) ==> F.]
20		wex 129	. . . . . 6 |- E.:((al -> *) -> *)
21	1	wl 59	. . . . . 6 |- \x:al A:(al -> *)
22	20, 21	wc 45	. . . . 5 |- (E.\x:al A):*
23	22	notval 135	. . . 4 |- T. |= [(~ (E.\x:al A)) = [(E.\x:al A) ==> F.]]
24	6, 23	a1i 28	. . 3 |- (A.\x:al (~ A)) |= [(~ (E.\x:al A)) = [(E.\x:al A) ==> F.]]
25	19, 24	mpbir 77	. 2 |- (A.\x:al (~ A)) |= (~ (E.\x:al A))
26	1	19.8a 160	. . . . . 6 |- A |= (E.\x:al A)
27		wtru 40	. . . . . 6 |- T.:*
28	26, 27	adantl 51	. . . . 5 |- (T., A) |= (E.\x:al A)
29	28	con3d 152	. . . 4 |- (T., (~ (E.\x:al A))) |= (~ A)
30	29	trul 37	. . 3 |- (~ (E.\x:al A)) |= (~ A)
31	3, 13	ax-17 95	. . . 4 |- T. |= [(\x:al ~ y:al) = ~ ]
32	20, 13	ax-17 95	. . . . 5 |- T. |= [(\x:al E.y:al) = E.]
33	1, 13	ax-hbl1 93	. . . . 5 |- T. |= [(\x:al \x:al Ay:al) = \x:al A]
34	20, 21, 13, 32, 33	hbc 100	. . . 4 |- T. |= [(\x:al (E.\x:al A)y:al) = (E.\x:al A)]
35	3, 22, 13, 31, 34	hbc 100	. . 3 |- T. |= [(\x:al (~ (E.\x:al A))y:al) = (~ (E.\x:al A))]
36	30, 35	alrimi 170	. 2 |- (~ (E.\x:al A)) |= (A.\x:al (~ A))
37	25, 36	dedi 75	1 |- T. |= [(A.\x:al (~ A)) = (~ (E.\x:al A))]

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  ~ tne 110   ==> tim 111  A.tal 112  E.tex 113

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-eta 165

This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120  df-ex 121

This theorem is referenced by:  exnal1  175  exnal  188  ax9  199