Theorem ax7 196

Description: Axiom of Quantifier Commutation. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint).

Hypothesis

Ref	Expression
ax7.1	|- R:*

Assertion

Ref	Expression
ax7	|- T. |= [(A.\x:al (A.\y:al R)) ==> (A.\y:al (A.\x:al R))]

Distinct variable group:   x,y,al

Proof of Theorem ax7

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wal 124	. . . . . . . 8 |- A.:((al -> *) -> *)
2		ax7.1	. . . . . . . . 9 |- R:*
3	2	wl 59	. . . . . . . 8 |- \y:al R:(al -> *)
4	1, 3	wc 45	. . . . . . 7 |- (A.\y:al R):*
5	4	ax4 140	. . . . . 6 |- (A.\x:al (A.\y:al R)) |= (A.\y:al R)
6	2	ax4 140	. . . . . 6 |- (A.\y:al R) |= R
7	5, 6	syl 16	. . . . 5 |- (A.\x:al (A.\y:al R)) |= R
8	4	wl 59	. . . . . 6 |- \x:al (A.\y:al R):(al -> *)
9		wv 58	. . . . . 6 |- z:al:al
10	1, 9	ax-17 95	. . . . . 6 |- T. |= [(\x:al A.z:al) = A.]
11	4, 9	ax-hbl1 93	. . . . . 6 |- T. |= [(\x:al \x:al (A.\y:al R)z:al) = \x:al (A.\y:al R)]
12	1, 8, 9, 10, 11	hbc 100	. . . . 5 |- T. |= [(\x:al (A.\x:al (A.\y:al R))z:al) = (A.\x:al (A.\y:al R))]
13	7, 12	alrimi 170	. . . 4 |- (A.\x:al (A.\y:al R)) |= (A.\x:al R)
14	1, 9	ax-17 95	. . . . 5 |- T. |= [(\y:al A.z:al) = A.]
15	2, 9	ax-hbl1 93	. . . . . . 7 |- T. |= [(\y:al \y:al Rz:al) = \y:al R]
16	1, 3, 9, 14, 15	hbc 100	. . . . . 6 |- T. |= [(\y:al (A.\y:al R)z:al) = (A.\y:al R)]
17	4, 9, 16	hbl 102	. . . . 5 |- T. |= [(\y:al \x:al (A.\y:al R)z:al) = \x:al (A.\y:al R)]
18	1, 8, 9, 14, 17	hbc 100	. . . 4 |- T. |= [(\y:al (A.\x:al (A.\y:al R))z:al) = (A.\x:al (A.\y:al R))]
19	13, 18	alrimi 170	. . 3 |- (A.\x:al (A.\y:al R)) |= (A.\y:al (A.\x:al R))
20		wtru 40	. . 3 |- T.:*
21	19, 20	adantl 51	. 2 |- (T., (A.\x:al (A.\y:al R))) |= (A.\y:al (A.\x:al R))
22	21	ex 148	1 |- T. |= [(A.\x:al (A.\y:al R)) ==> (A.\y: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   ==> tim 111  A.tal 112

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-an 118  df-im 119

This theorem is referenced by: (None)