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Theorem ecase 153
Description: Elimination by cases.
Hypotheses
Ref Expression
ecase.1 |- A:*
ecase.2 |- B:*
ecase.3 |- T:*
ecase.4 |- R |= [A \/ B]
ecase.5 |- (R, A) |= T
ecase.6 |- (R, B) |= T
Assertion
Ref Expression
ecase |- R |= T
Proof of Theorem ecase
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ecase.3 . 2 |- T:*
2 ecase.6 . . 3 |- (R, B) |= T
32ex 148 . 2 |- R |= [B ==> T]
4 wim 127 . . . 4 |- ==> :(* -> (* -> *))
5 ecase.2 . . . . 5 |- B:*
64, 5, 1wov 64 . . . 4 |- [B ==> T]:*
74, 6, 1wov 64 . . 3 |- [[B ==> T] ==> T]:*
8 ecase.5 . . . 4 |- (R, A) |= T
98ex 148 . . 3 |- R |= [A ==> T]
10 ecase.4 . . . . 5 |- R |= [A \/ B]
1110ax-cb1 29 . . . . . 6 |- R:*
12 ecase.1 . . . . . . 7 |- A:*
1312, 5orval 137 . . . . . 6 |- T. |= [[A \/ B] = (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])]
1411, 13a1i 28 . . . . 5 |- R |= [[A \/ B] = (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])]
1510, 14mpbi 72 . . . 4 |- R |= (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]])
16 wv 58 . . . . . . 7 |- x:*:*
174, 12, 16wov 64 . . . . . 6 |- [A ==> x:*]:*
184, 5, 16wov 64 . . . . . . 7 |- [B ==> x:*]:*
194, 18, 16wov 64 . . . . . 6 |- [[B ==> x:*] ==> x:*]:*
204, 17, 19wov 64 . . . . 5 |- [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]:*
2116, 1weqi 68 . . . . . . . 8 |- [x:* = T]:*
2221id 25 . . . . . . 7 |- [x:* = T] |= [x:* = T]
234, 12, 16, 22oveq2 91 . . . . . 6 |- [x:* = T] |= [[A ==> x:*] = [A ==> T]]
244, 5, 16, 22oveq2 91 . . . . . . 7 |- [x:* = T] |= [[B ==> x:*] = [B ==> T]]
254, 18, 16, 24, 22oveq12 90 . . . . . 6 |- [x:* = T] |= [[[B ==> x:*] ==> x:*] = [[B ==> T] ==> T]]
264, 17, 19, 23, 25oveq12 90 . . . . 5 |- [x:* = T] |= [[[A ==> x:*] ==> [[B ==> x:*] ==> x:*]] = [[A ==> T] ==> [[B ==> T] ==> T]]]
2720, 1, 26cla4v 142 . . . 4 |- (A.\x:* [[A ==> x:*] ==> [[B ==> x:*] ==> x:*]]) |= [[A ==> T] ==> [[B ==> T] ==> T]]
2815, 27syl 16 . . 3 |- R |= [[A ==> T] ==> [[B ==> T] ==> T]]
297, 9, 28mpd 146 . 2 |- R |= [[B ==> T] ==> T]
301, 3, 29mpd 146 1 |- R |= T
Colors of variables: type var term
Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [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  notnot  187  ax3  192
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