Theorem exnal1 187

Description: Forward direction of exnal 201. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypothesis

Ref	Expression
alnex1.1	|- A:*

Assertion

Ref	Expression
exnal1	|- (E.\x:al (~ A)) |= (~ (A.\x:al A))

Distinct variable group:   al,x

Proof of Theorem exnal1

Step	Hyp	Ref	Expression
1		wex 139	. . . 4 |- E.:((al -> *) -> *)
2		wnot 138	. . . . . 6 |- ~ :(* -> *)
3		alnex1.1	. . . . . 6 |- A:*
4	2, 3	wc 50	. . . . 5 |- (~ A):*
5	4	wl 66	. . . 4 |- \x:al (~ A):(al -> *)
6	1, 5	wc 50	. . 3 |- (E.\x:al (~ A)):*
7	3	notnot1 160	. . . . . 6 |- A |= (~ (~ A))
8		wtru 43	. . . . . 6 |- T.:*
9	7, 8	adantl 56	. . . . 5 |- (T., A) |= (~ (~ A))
10	9	alimdv 184	. . . 4 |- (T., (A.\x:al A)) |= (A.\x:al (~ (~ A)))
11		wal 134	. . . . . . 7 |- A.:((al -> *) -> *)
12	2, 4	wc 50	. . . . . . . 8 |- (~ (~ A)):*
13	12	wl 66	. . . . . . 7 |- \x:al (~ (~ A)):(al -> *)
14	11, 13	wc 50	. . . . . 6 |- (A.\x:al (~ (~ A))):*
15	14	id 25	. . . . 5 |- (A.\x:al (~ (~ A))) |= (A.\x:al (~ (~ A)))
16	4	alnex 186	. . . . . 6 |- T. |= [(A.\x:al (~ (~ A))) = (~ (E.\x:al (~ A)))]
17	14, 16	a1i 28	. . . . 5 |- (A.\x:al (~ (~ A))) |= [(A.\x:al (~ (~ A))) = (~ (E.\x:al (~ A)))]
18	15, 17	mpbi 82	. . . 4 |- (A.\x:al (~ (~ A))) |= (~ (E.\x:al (~ A)))
19	10, 18	syl 16	. . 3 |- (T., (A.\x:al A)) |= (~ (E.\x:al (~ A)))
20	6, 19	con2d 161	. 2 |- (T., (E.\x:al (~ A))) |= (~ (A.\x:al A))
21	20	trul 39	1 |- (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  kct 10   |= 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

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: (None)