Theorem ax7 209

Description: Axiom of Quantifier Commutation. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by Mario Carneiro, 10-Oct-2014.)

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 134	. . . . . . . 8 |- A.:((al -> *) -> *)
2		ax7.1	. . . . . . . . 9 |- R:*
3	2	wl 66	. . . . . . . 8 |- \y:al R:(al -> *)
4	1, 3	wc 50	. . . . . . 7 |- (A.\y:al R):*
5	4	ax4 150	. . . . . 6 |- (A.\x:al (A.\y:al R)) |= (A.\y:al R)
6	2	ax4 150	. . . . . 6 |- (A.\y:al R) |= R
7	5, 6	syl 16	. . . . 5 |- (A.\x:al (A.\y:al R)) |= R
8	4	wl 66	. . . . . 6 |- \x:al (A.\y:al R):(al -> *)
9		wv 64	. . . . . 6 |- z:al:al
10	1, 9	ax-17 105	. . . . . 6 |- T. |= [(\x:al A.z:al) = A.]
11	4, 9	ax-hbl1 103	. . . . . 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 110	. . . . 5 |- T. |= [(\x:al (A.\x:al (A.\y:al R))z:al) = (A.\x:al (A.\y:al R))]
13	7, 12	alrimi 182	. . . 4 |- (A.\x:al (A.\y:al R)) |= (A.\x:al R)
14	1, 9	ax-17 105	. . . . 5 |- T. |= [(\y:al A.z:al) = A.]
15	2, 9	ax-hbl1 103	. . . . . . 7 |- T. |= [(\y:al \y:al Rz:al) = \y:al R]
16	1, 3, 9, 14, 15	hbc 110	. . . . . 6 |- T. |= [(\y:al (A.\y:al R)z:al) = (A.\y:al R)]
17	4, 9, 16	hbl 112	. . . . 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 110	. . . 4 |- T. |= [(\y:al (A.\x:al (A.\y:al R))z:al) = (A.\x:al (A.\y:al R))]
19	13, 18	alrimi 182	. . 3 |- (A.\x:al (A.\y:al R)) |= (A.\y:al (A.\x:al R))
20		wtru 43	. . 3 |- T.:*
21	19, 20	adantl 56	. 2 |- (T., (A.\x:al (A.\y:al R))) |= (A.\y:al (A.\x:al R))
22	21	ex 158	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 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-an 128  df-im 129

This theorem is referenced by: (None)