Theorem exnal 201

Description: Theorem 19.14 of [Margaris] p. 90. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypothesis

Ref	Expression
exmid.1	|- A:*

Assertion

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

Distinct variable groups:   x,A   al,x

Proof of Theorem exnal

Step	Hyp	Ref	Expression
1		wnot 138	. . 3 |- ~ :(* -> *)
2		wex 139	. . . . 5 |- E.:((al -> *) -> *)
3		exmid.1	. . . . . . 7 |- A:*
4	1, 3	wc 50	. . . . . 6 |- (~ A):*
5	4	wl 66	. . . . 5 |- \x:al (~ A):(al -> *)
6	2, 5	wc 50	. . . 4 |- (E.\x:al (~ A)):*
7	1, 6	wc 50	. . 3 |- (~ (E.\x:al (~ A))):*
8	1, 7	wc 50	. 2 |- (~ (~ (E.\x:al (~ A)))):*
9		wal 134	. . . . 5 |- A.:((al -> *) -> *)
10	1, 4	wc 50	. . . . . 6 |- (~ (~ A)):*
11	10	wl 66	. . . . 5 |- \x:al (~ (~ A)):(al -> *)
12	9, 11	wc 50	. . . 4 |- (A.\x:al (~ (~ A))):*
13	4	alnex 186	. . . 4 |- T. |= [(A.\x:al (~ (~ A))) = (~ (E.\x:al (~ A)))]
14	12, 13	eqcomi 79	. . 3 |- T. |= [(~ (E.\x:al (~ A))) = (A.\x:al (~ (~ A)))]
15	1, 7, 14	ceq2 90	. 2 |- T. |= [(~ (~ (E.\x:al (~ A)))) = (~ (A.\x:al (~ (~ A))))]
16	6	notnot 200	. 2 |- T. |= [(E.\x:al (~ A)) = (~ (~ (E.\x:al (~ A))))]
17	3	wl 66	. . . 4 |- \x:al A:(al -> *)
18	9, 17	wc 50	. . 3 |- (A.\x:al A):*
19	3	notnot 200	. . . . 5 |- T. |= [A = (~ (~ A))]
20	3, 19	leq 91	. . . 4 |- T. |= [\x:al A = \x:al (~ (~ A))]
21	9, 17, 20	ceq2 90	. . 3 |- T. |= [(A.\x:al A) = (A.\x:al (~ (~ A)))]
22	1, 18, 21	ceq2 90	. 2 |- T. |= [(~ (A.\x:al A)) = (~ (A.\x:al (~ (~ A))))]
23	8, 15, 16, 22	3eqtr4i 96	1 |- T. |= [(E.\x:al (~ A)) = (~ (A.\x:al A))]

Syntax hints:   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  ~ tne 120  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  ax-wat 192  ax-ac 196

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  df-or 132

This theorem is referenced by: (None)