Theorem ax6 208

Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypothesis

Ref	Expression
ax6.1	|- R:*

Assertion

Ref	Expression
ax6	|- T. |= [(~ (A.\x:al R)) ==> (A.\x:al (~ (A.\x:al R)))]

Distinct variable group:   al,x

Proof of Theorem ax6

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wnot 138	. . 3 |- ~ :(* -> *)
2		wal 134	. . . 4 |- A.:((al -> *) -> *)
3		ax6.1	. . . . 5 |- R:*
4	3	wl 66	. . . 4 |- \x:al R:(al -> *)
5	2, 4	wc 50	. . 3 |- (A.\x:al R):*
6	1, 5	wc 50	. 2 |- (~ (A.\x:al R)):*
7		wv 64	. . 3 |- y:al:al
8	1, 7	ax-17 105	. . 3 |- T. |= [(\x:al ~ y:al) = ~ ]
9	2, 7	ax-17 105	. . . 4 |- T. |= [(\x:al A.y:al) = A.]
10	3, 7	ax-hbl1 103	. . . 4 |- T. |= [(\x:al \x:al Ry:al) = \x:al R]
11	2, 4, 7, 9, 10	hbc 110	. . 3 |- T. |= [(\x:al (A.\x:al R)y:al) = (A.\x:al R)]
12	1, 5, 7, 8, 11	hbc 110	. 2 |- T. |= [(\x:al (~ (A.\x:al R))y:al) = (~ (A.\x:al R))]
13	6, 12	isfree 188	1 |- T. |= [(~ (A.\x:al R)) ==> (A.\x:al (~ (A.\x:al R)))]

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  ~ tne 120   ==> tim 121  A.tal 122

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

This theorem is referenced by: (None)