Theorem ax5 207

Description: Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypotheses

Ref	Expression
ax5.1	|- R:*
ax5.2	|- S:*

Assertion

Ref	Expression
ax5	|- T. |= [(A.\x:al [R ==> S]) ==> [(A.\x:al R) ==> (A.\x:al S)]]

Distinct variable group:   al,x

Proof of Theorem ax5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax5.2	. . . . . 6 |- S:*
2		ax5.1	. . . . . . . 8 |- R:*
3	2	ax4 150	. . . . . . 7 |- (A.\x:al R) |= R
4		wal 134	. . . . . . . 8 |- A.:((al -> *) -> *)
5		wim 137	. . . . . . . . . 10 |- ==> :(* -> (* -> *))
6	5, 2, 1	wov 72	. . . . . . . . 9 |- [R ==> S]:*
7	6	wl 66	. . . . . . . 8 |- \x:al [R ==> S]:(al -> *)
8	4, 7	wc 50	. . . . . . 7 |- (A.\x:al [R ==> S]):*
9	3, 8	adantl 56	. . . . . 6 |- ((A.\x:al [R ==> S]), (A.\x:al R)) |= R
10	6	ax4 150	. . . . . . 7 |- (A.\x:al [R ==> S]) |= [R ==> S]
11	3	ax-cb1 29	. . . . . . 7 |- (A.\x:al R):*
12	10, 11	adantr 55	. . . . . 6 |- ((A.\x:al [R ==> S]), (A.\x:al R)) |= [R ==> S]
13	1, 9, 12	mpd 156	. . . . 5 |- ((A.\x:al [R ==> S]), (A.\x:al R)) |= S
14		wv 64	. . . . . 6 |- y:al:al
15	4, 14	ax-17 105	. . . . . . 7 |- T. |= [(\x:al A.y:al) = A.]
16	6, 14	ax-hbl1 103	. . . . . . 7 |- T. |= [(\x:al \x:al [R ==> S]y:al) = \x:al [R ==> S]]
17	4, 7, 14, 15, 16	hbc 110	. . . . . 6 |- T. |= [(\x:al (A.\x:al [R ==> S])y:al) = (A.\x:al [R ==> S])]
18	2	wl 66	. . . . . . 7 |- \x:al R:(al -> *)
19	2, 14	ax-hbl1 103	. . . . . . 7 |- T. |= [(\x:al \x:al Ry:al) = \x:al R]
20	4, 18, 14, 15, 19	hbc 110	. . . . . 6 |- T. |= [(\x:al (A.\x:al R)y:al) = (A.\x:al R)]
21	8, 14, 11, 17, 20	hbct 155	. . . . 5 |- T. |= [(\x:al ((A.\x:al [R ==> S]), (A.\x:al R))y:al) = ((A.\x:al [R ==> S]), (A.\x:al R))]
22	13, 21	alrimi 182	. . . 4 |- ((A.\x:al [R ==> S]), (A.\x:al R)) |= (A.\x:al S)
23	22	ex 158	. . 3 |- (A.\x:al [R ==> S]) |= [(A.\x:al R) ==> (A.\x:al S)]
24		wtru 43	. . 3 |- T.:*
25	23, 24	adantl 56	. 2 |- (T., (A.\x:al [R ==> S])) |= [(A.\x:al R) ==> (A.\x:al S)]
26	25	ex 158	1 |- T. |= [(A.\x:al [R ==> S]) ==> [(A.\x:al R) ==> (A.\x:al S)]]

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  [kbr 9  kct 10   |= 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:  ax11  214