Theorem olc 154

Description: Or introduction.

Hypotheses

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

Assertion

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

Proof of Theorem olc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wim 127	. . . 4 |- ==> :(* -> (* -> *))
2		olc.1	. . . . 5 |- A:*
3		wv 58	. . . . 5 |- x:*:*
4	1, 2, 3	wov 64	. . . 4 |- [A ==> x:*]:*
5		olc.2	. . . . . 6 |- B:*
6	1, 5, 3	wov 64	. . . . 5 |- [B ==> x:*]:*
7	1, 6, 3	wov 64	. . . 4 |- [[B ==> x:*] ==> x:*]:*
8	1, 4, 7	wov 64	. . 3 |- [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:*
9		wtru 40	. . . 4 |- T.:*
10	5, 6	simpl 22	. . . . . . . . 9 |- (B, [B ==> x:*]) |= B
11	5, 6	simpr 23	. . . . . . . . 9 |- (B, [B ==> x:*]) |= [B ==> x:*]
12	3, 10, 11	mpd 146	. . . . . . . 8 |- (B, [B ==> x:*]) |= x:*
13	12	ex 148	. . . . . . 7 |- B |= [[B ==> x:*] ==> x:*]
14	13, 4	adantr 50	. . . . . 6 |- (B, [A ==> x:*]) |= [[B ==> x:*] ==> x:*]
15	14	ex 148	. . . . 5 |- B |= [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]
16	15	eqtru 76	. . . 4 |- B |= [T. = [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]]
17	9, 16	eqcomi 70	. . 3 |- B |= [[[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = T.]
18	8, 17	leq 81	. 2 |- B |= [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]
19		wor 130	. . . . 5 |- \/ :(* -> (* -> *))
20	19, 2, 5	wov 64	. . . 4 |- [A \/ B]:*
21	2, 5	orval 137	. . . 4 |- T. |= [[A \/ B] = (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])]
22	8	wl 59	. . . . 5 |- \x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:(* -> *)
23	22	alval 132	. . . 4 |- T. |= [(A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]) = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
24	20, 21, 23	eqtri 85	. . 3 |- T. |= [[A \/ B] = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
25	5, 24	a1i 28	. 2 |- B |= [[A \/ B] = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
26	18, 25	mpbir 77	1 |- B |= [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 111  A.tal 112   \/ tor 114

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

This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119  df-or 122

This theorem is referenced by:  exmid  186