Theorem ax9 212

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. (Contributed by Mario Carneiro, 10-Oct-2014.)

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 64	. . . . . 6 |- x:al:al
2		ax9.1	. . . . . 6 |- A:al
3	1, 2	weqi 76	. . . . 5 |- [x:al = A]:*
4	3	19.8a 170	. . . 4 |- [x:al = A] |= (E.\x:al [x:al = A])
5		wex 139	. . . . 5 |- E.:((al -> *) -> *)
6	3	wl 66	. . . . 5 |- \x:al [x:al = A]:(al -> *)
7		wv 64	. . . . 5 |- y:al:al
8	5, 7	ax-17 105	. . . . 5 |- T. |= [(\x:al E.y:al) = E.]
9	3, 7	ax-hbl1 103	. . . . 5 |- T. |= [(\x:al \x:al [x:al = A]y:al) = \x:al [x:al = A]]
10	5, 6, 7, 8, 9	hbc 110	. . . 4 |- T. |= [(\x:al (E.\x:al [x:al = A])y:al) = (E.\x:al [x:al = A])]
11		wtru 43	. . . . 5 |- T.:*
12	11, 7	ax-17 105	. . . 4 |- T. |= [(\x:al T.y:al) = T.]
13	5, 6	wc 50	. . . . 5 |- (E.\x:al [x:al = A]):*
14	3, 13	eqid 83	. . . 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 86	. . . . 5 |- [x:al = A] |= [T. = [x:al = A]]
17	11, 16	eqcomi 79	. . . 4 |- [x:al = A] |= [[x:al = A] = T.]
18	4, 10, 12, 14, 17	ax-inst 113	. . 3 |- T. |= (E.\x:al [x:al = A])
19	13	notnot1 160	. . 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 138	. . 3 |- ~ :(* -> *)
22		wal 134	. . . 4 |- A.:((al -> *) -> *)
23	21, 3	wc 50	. . . . 5 |- (~ [x:al = A]):*
24	23	wl 66	. . . 4 |- \x:al (~ [x:al = A]):(al -> *)
25	22, 24	wc 50	. . 3 |- (A.\x:al (~ [x:al = A])):*
26	3	alnex 186	. . 3 |- T. |= [(A.\x:al (~ [x:al = A])) = (~ (E.\x:al [x:al = A]))]
27	21, 25, 26	ceq2 90	. 2 |- T. |= [(~ (A.\x:al (~ [x:al = A]))) = (~ (~ (E.\x:al [x:al = A])))]
28	20, 27	mpbir 87	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 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)