Theorem alnex 186

Description: Theorem 19.7 of [Margaris] p. 89. (Contributed by Mario Carneiro, 10-Oct-2014.)

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 135	. . . . . 6 |- F.:*
3		wnot 138	. . . . . . . . 9 |- ~ :(* -> *)
4	3, 1	wc 50	. . . . . . . 8 |- (~ A):*
5	4	ax4 150	. . . . . . 7 |- (A.\x:al (~ A)) |= (~ A)
6	5	ax-cb1 29	. . . . . . . 8 |- (A.\x:al (~ A)):*
7	1	notval 145	. . . . . . . 8 |- T. |= [(~ A) = [A ==> F.]]
8	6, 7	a1i 28	. . . . . . 7 |- (A.\x:al (~ A)) |= [(~ A) = [A ==> F.]]
9	5, 8	mpbi 82	. . . . . 6 |- (A.\x:al (~ A)) |= [A ==> F.]
10	1, 2, 9	imp 157	. . . . 5 |- ((A.\x:al (~ A)), A) |= F.
11		wal 134	. . . . . 6 |- A.:((al -> *) -> *)
12	4	wl 66	. . . . . 6 |- \x:al (~ A):(al -> *)
13		wv 64	. . . . . 6 |- y:al:al
14	11, 13	ax-17 105	. . . . . 6 |- T. |= [(\x:al A.y:al) = A.]
15	4, 13	ax-hbl1 103	. . . . . 6 |- T. |= [(\x:al \x:al (~ A)y:al) = \x:al (~ A)]
16	11, 12, 13, 14, 15	hbc 110	. . . . 5 |- T. |= [(\x:al (A.\x:al (~ A))y:al) = (A.\x:al (~ A))]
17	2, 13	ax-17 105	. . . . 5 |- T. |= [(\x:al F.y:al) = F.]
18	10, 16, 17	exlimd 183	. . . 4 |- ((A.\x:al (~ A)), (E.\x:al A)) |= F.
19	18	ex 158	. . 3 |- (A.\x:al (~ A)) |= [(E.\x:al A) ==> F.]
20		wex 139	. . . . . 6 |- E.:((al -> *) -> *)
21	1	wl 66	. . . . . 6 |- \x:al A:(al -> *)
22	20, 21	wc 50	. . . . 5 |- (E.\x:al A):*
23	22	notval 145	. . . 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 87	. 2 |- (A.\x:al (~ A)) |= (~ (E.\x:al A))
26	1	19.8a 170	. . . . . 6 |- A |= (E.\x:al A)
27		wtru 43	. . . . . 6 |- T.:*
28	26, 27	adantl 56	. . . . 5 |- (T., A) |= (E.\x:al A)
29	28	con3d 162	. . . 4 |- (T., (~ (E.\x:al A))) |= (~ A)
30	29	trul 39	. . 3 |- (~ (E.\x:al A)) |= (~ A)
31	3, 13	ax-17 105	. . . 4 |- T. |= [(\x:al ~ y:al) = ~ ]
32	20, 13	ax-17 105	. . . . 5 |- T. |= [(\x:al E.y:al) = E.]
33	1, 13	ax-hbl1 103	. . . . 5 |- T. |= [(\x:al \x:al Ay:al) = \x:al A]
34	20, 21, 13, 32, 33	hbc 110	. . . 4 |- T. |= [(\x:al (E.\x:al A)y:al) = (E.\x:al A)]
35	3, 22, 13, 31, 34	hbc 110	. . 3 |- T. |= [(\x:al (~ (E.\x:al A))y:al) = (~ (E.\x:al A))]
36	30, 35	alrimi 182	. 2 |- (~ (E.\x:al A)) |= (A.\x:al (~ A))
37	25, 36	dedi 85	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 118  ~ tne 120   ==> 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-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-eta 177

This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130  df-ex 131

This theorem is referenced by:  exnal1  187  exnal  201  ax9  212