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Theorem ecase 163
 Description: Elimination by cases. (Contributed by Mario Carneiro, 9-Oct-2014.)
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 RT

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 158 . 2 R⊧[BT]
4 wim 137 . . . 4 ⇒ :(∗ → (∗ → ∗))
5 ecase.2 . . . . 5 B:∗
64, 5, 1wov 72 . . . 4 [BT]:∗
74, 6, 1wov 72 . . 3 [[BT] ⇒ T]:∗
8 ecase.5 . . . 4 (R, A)⊧T
98ex 158 . . 3 R⊧[AT]
10 ecase.4 . . . . 5 R⊧[A B]
1110ax-cb1 29 . . . . . 6 R:∗
12 ecase.1 . . . . . . 7 A:∗
1312, 5orval 147 . . . . . 6 ⊤⊧[[A B] = (λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]])]
1411, 13a1i 28 . . . . 5 R⊧[[A B] = (λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]])]
1510, 14mpbi 82 . . . 4 R⊧(λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]])
16 wv 64 . . . . . . 7 x:∗:∗
174, 12, 16wov 72 . . . . . 6 [Ax:∗]:∗
184, 5, 16wov 72 . . . . . . 7 [Bx:∗]:∗
194, 18, 16wov 72 . . . . . 6 [[Bx:∗] ⇒ x:∗]:∗
204, 17, 19wov 72 . . . . 5 [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]]:∗
2116, 1weqi 76 . . . . . . . 8 [x:∗ = T]:∗
2221id 25 . . . . . . 7 [x:∗ = T]⊧[x:∗ = T]
234, 12, 16, 22oveq2 101 . . . . . 6 [x:∗ = T]⊧[[Ax:∗] = [AT]]
244, 5, 16, 22oveq2 101 . . . . . . 7 [x:∗ = T]⊧[[Bx:∗] = [BT]]
254, 18, 16, 24, 22oveq12 100 . . . . . 6 [x:∗ = T]⊧[[[Bx:∗] ⇒ x:∗] = [[BT] ⇒ T]]
264, 17, 19, 23, 25oveq12 100 . . . . 5 [x:∗ = T]⊧[[[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]] = [[AT] ⇒ [[BT] ⇒ T]]]
2720, 1, 26cla4v 152 . . . 4 (λx:∗ [[Ax:∗] ⇒ [[Bx:∗] ⇒ x:∗]])⊧[[AT] ⇒ [[BT] ⇒ T]]
2815, 27syl 16 . . 3 R⊧[[AT] ⇒ [[BT] ⇒ T]]
297, 9, 28mpd 156 . 2 R⊧[[BT] ⇒ T]
301, 3, 29mpd 156 1 RT
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12   ⇒ tim 121  ∀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  notnot  200  ax3  205
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