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Theorem imval 146
 Description: Value of the implication. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
imval.1 A:∗
imval.2 B:∗
Assertion
Ref Expression
imval ⊤⊧[[AB] = [[A B] = A]]

Proof of Theorem imval
Dummy variables p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wim 137 . . 3 ⇒ :(∗ → (∗ → ∗))
2 imval.1 . . 3 A:∗
3 imval.2 . . 3 B:∗
41, 2, 3wov 72 . 2 [AB]:∗
5 df-im 129 . . 3 ⊤⊧[ ⇒ = λp:∗ λq:∗ [[p:∗ q:∗] = p:∗]]
61, 2, 3, 5oveq 102 . 2 ⊤⊧[[AB] = [Aλp:∗ λq:∗ [[p:∗ q:∗] = p:∗]B]]
7 wan 136 . . . . 5 :(∗ → (∗ → ∗))
8 wv 64 . . . . 5 p:∗:∗
9 wv 64 . . . . 5 q:∗:∗
107, 8, 9wov 72 . . . 4 [p:∗ q:∗]:∗
1110, 8weqi 76 . . 3 [[p:∗ q:∗] = p:∗]:∗
12 weq 41 . . . 4 = :(∗ → (∗ → ∗))
138, 2weqi 76 . . . . . 6 [p:∗ = A]:∗
1413id 25 . . . . 5 [p:∗ = A]⊧[p:∗ = A]
157, 8, 9, 14oveq1 99 . . . 4 [p:∗ = A]⊧[[p:∗ q:∗] = [A q:∗]]
1612, 10, 8, 15, 14oveq12 100 . . 3 [p:∗ = A]⊧[[[p:∗ q:∗] = p:∗] = [[A q:∗] = A]]
177, 2, 9wov 72 . . . 4 [A q:∗]:∗
189, 3weqi 76 . . . . . 6 [q:∗ = B]:∗
1918id 25 . . . . 5 [q:∗ = B]⊧[q:∗ = B]
207, 2, 9, 19oveq2 101 . . . 4 [q:∗ = B]⊧[[A q:∗] = [A B]]
2112, 17, 2, 20oveq1 99 . . 3 [q:∗ = B]⊧[[[A q:∗] = A] = [[A B] = A]]
2211, 2, 3, 16, 21ovl 117 . 2 ⊤⊧[[Aλp:∗ λq:∗ [[p:∗ q:∗] = p:∗]B] = [[A B] = A]]
234, 6, 22eqtri 95 1 ⊤⊧[[AB] = [[A B] = A]]
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ∧ tan 119   ⇒ tim 121 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-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-an 128  df-im 129 This theorem is referenced by:  mpd  156  ex  158
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