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Theorem orc 155
Description: Or introduction.
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.
StepHypRef Expression
1 wim 127 . . . 4 |- ==> :(* -> (* -> *))
2 olc.1 . . . . 5 |- A:*
3 wv 58 . . . . 5 |- x:*:*
41, 2, 3wov 64 . . . 4 |- [A ==> x:*]:*
5 olc.2 . . . . . 6 |- B:*
61, 5, 3wov 64 . . . . 5 |- [B ==> x:*]:*
71, 6, 3wov 64 . . . 4 |- [[B ==> x:*] ==> x:*]:*
81, 4, 7wov 64 . . 3 |- [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:*
9 wtru 40 . . . 4 |- T.:*
102, 4simpl 22 . . . . . . . . 9 |- (A, [A ==> x:*]) |= A
112, 4simpr 23 . . . . . . . . 9 |- (A, [A ==> x:*]) |= [A ==> x:*]
123, 10, 11mpd 146 . . . . . . . 8 |- (A, [A ==> x:*]) |= x:*
1312, 6adantr 50 . . . . . . 7 |- ((A, [A ==> x:*]), [B ==> x:*]) |= x:*
1413ex 148 . . . . . 6 |- (A, [A ==> x:*]) |= [[B ==> x:*] ==> x:*]
1514ex 148 . . . . 5 |- A |= [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]
1615eqtru 76 . . . 4 |- A |= [T. = [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]]
179, 16eqcomi 70 . . 3 |- A |= [[[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = T.]
188, 17leq 81 . 2 |- A |= [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]
19 wor 130 . . . . 5 |- \/ :(* -> (* -> *))
2019, 2, 5wov 64 . . . 4 |- [A \/ B]:*
212, 5orval 137 . . . 4 |- T. |= [[A \/ B] = (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])]
228wl 59 . . . . 5 |- \x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:(* -> *)
2322alval 132 . . . 4 |- T. |= [(A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]) = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
2420, 21, 23eqtri 85 . . 3 |- T. |= [[A \/ B] = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
252, 24a1i 28 . 2 |- A |= [[A \/ B] = [\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = \x:* T.]]
2618, 25mpbir 77 1 |- A |= [A \/ B]
Colors of variables: type var term
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
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