Theorem orc 165

Description: Or introduction. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypotheses

Ref	Expression
olc.1	|- A:*
olc.2	|- B:*

Assertion

Ref	Expression
orc	|- A |= [A \/ B]

Proof of Theorem orc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wim 137	. . . 4 |- ==> :(* -> (* -> *))
2		olc.1	. . . . 5 |- A:*
3		wv 64	. . . . 5 |- x:*:*
4	1, 2, 3	wov 72	. . . 4 |- [A ==> x:*]:*
5		olc.2	. . . . . 6 |- B:*
6	1, 5, 3	wov 72	. . . . 5 |- [B ==> x:*]:*
7	1, 6, 3	wov 72	. . . 4 |- [[B ==> x:*] ==> x:*]:*
8	1, 4, 7	wov 72	. . 3 |- [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:*
9		wtru 43	. . . 4 |- T.:*
10	2, 4	simpl 22	. . . . . . . . 9 |- (A, [A ==> x:*]) |= A
11	2, 4	simpr 23	. . . . . . . . 9 |- (A, [A ==> x:*]) |= [A ==> x:*]
12	3, 10, 11	mpd 156	. . . . . . . 8 |- (A, [A ==> x:*]) |= x:*
13	12, 6	adantr 55	. . . . . . 7 |- ((A, [A ==> x:*]), [B ==> x:*]) |= x:*
14	13	ex 158	. . . . . 6 |- (A, [A ==> x:*]) |= [[B ==> x:*] ==> x:*]
15	14	ex 158	. . . . 5 |- A |= [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]
16	15	eqtru 86	. . . 4 |- A |= [T. = [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]]
17	9, 16	eqcomi 79	. . 3 |- A |= [[[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = T.]
18	8, 17	leq 91	. 2 |- A |= [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]
19		wor 140	. . . . 5 |- \/ :(* -> (* -> *))
20	19, 2, 5	wov 72	. . . 4 |- [A \/ B]:*
21	2, 5	orval 147	. . . 4 |- T. |= [[A \/ B] = (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])]
22	8	wl 66	. . . . 5 |- \x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:(* -> *)
23	22	alval 142	. . . 4 |- T. |= [(A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]) = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
24	20, 21, 23	eqtri 95	. . 3 |- T. |= [[A \/ B] = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
25	2, 24	a1i 28	. 2 |- A |= [[A \/ B] = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
26	18, 25	mpbir 87	1 |- A |= [A \/ B]

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 121  A.tal 122   \/ tor 124

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

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129  df-or 132

This theorem is referenced by:  exmid  199