Theorem ax9 199

Description: Axiom of Equality. Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Hypothesis

Ref	Expression
ax9.1	|- A:al

Assertion

Ref	Expression
ax9	|- T. |= (~ (A.\x:al (~ [x:al = A])))

Distinct variable group:   al,x

Proof of Theorem ax9

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wv 58	. . . . . 6 |- x:al:al
2		ax9.1	. . . . . 6 |- A:al
3	1, 2	weqi 68	. . . . 5 |- [x:al = A]:*
4	3	19.8a 160	. . . 4 |- [x:al = A] |= (E.\x:al [x:al = A])
5		wex 129	. . . . 5 |- E.:((al -> *) -> *)
6	3	wl 59	. . . . 5 |- \x:al [x:al = A]:(al -> *)
7		wv 58	. . . . 5 |- y:al:al
8	5, 7	ax-17 95	. . . . 5 |- T. |= [(\x:al E.y:al) = E.]
9	3, 7	ax-hbl1 93	. . . . 5 |- T. |= [(\x:al \x:al [x:al = A]y:al) = \x:al [x:al = A]]
10	5, 6, 7, 8, 9	hbc 100	. . . 4 |- T. |= [(\x:al (E.\x:al [x:al = A])y:al) = (E.\x:al [x:al = A])]
11		wtru 40	. . . . 5 |- T.:*
12	11, 7	ax-17 95	. . . 4 |- T. |= [(\x:al T.y:al) = T.]
13	5, 6	wc 45	. . . . 5 |- (E.\x:al [x:al = A]):*
14	3, 13	eqid 73	. . . 4 |- [x:al = A] |= [(E.\x:al [x:al = A]) = (E.\x:al [x:al = A])]
15	3	id 25	. . . . . 6 |- [x:al = A] |= [x:al = A]
16	15	eqtru 76	. . . . 5 |- [x:al = A] |= [T. = [x:al = A]]
17	11, 16	eqcomi 70	. . . 4 |- [x:al = A] |= [[x:al = A] = T.]
18	4, 10, 12, 14, 17	ax-inst 103	. . 3 |- T. |= (E.\x:al [x:al = A])
19	13	notnot1 150	. . 3 |- (E.\x:al [x:al = A]) |= (~ (~ (E.\x:al [x:al = A])))
20	18, 19	syl 16	. 2 |- T. |= (~ (~ (E.\x:al [x:al = A])))
21		wnot 128	. . 3 |- ~ :(* -> *)
22		wal 124	. . . 4 |- A.:((al -> *) -> *)
23	21, 3	wc 45	. . . . 5 |- (~ [x:al = A]):*
24	23	wl 59	. . . 4 |- \x:al (~ [x:al = A]):(al -> *)
25	22, 24	wc 45	. . 3 |- (A.\x:al (~ [x:al = A])):*
26	3	alnex 174	. . 3 |- T. |= [(A.\x:al (~ [x:al = A])) = (~ (E.\x:al [x:al = A]))]
27	21, 25, 26	ceq2 80	. 2 |- T. |= [(~ (A.\x:al (~ [x:al = A]))) = (~ (~ (E.\x:al [x:al = A])))]
28	20, 27	mpbir 77	1 |- T. |= (~ (A.\x:al (~ [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  ~ tne 110  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: (None)