Higher-Order Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HOLE Home  >  Th. List  >  hbct GIF version

Theorem hbct 155
 Description: Hypothesis builder for context conjunction. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
hbct.1 A:∗
hbct.2 B:α
hbct.3 C:∗
hbct.4 R⊧[(λx:α AB) = A]
hbct.5 R⊧[(λx:α CB) = C]
Assertion
Ref Expression
hbct R⊧[(λx:α (A, C)B) = (A, C)]

Proof of Theorem hbct
StepHypRef Expression
1 hbct.4 . . . 4 R⊧[(λx:α AB) = A]
21ax-cb1 29 . . 3 R:∗
32trud 27 . 2 R⊧⊤
4 hbct.1 . . . 4 A:∗
5 hbct.3 . . . 4 C:∗
64, 5wct 48 . . 3 (A, C):∗
7 hbct.2 . . 3 B:α
8 wan 136 . . . . 5 :(∗ → (∗ → ∗))
98, 4, 5wov 72 . . . 4 [A C]:∗
104, 5dfan2 154 . . . 4 ⊤⊧[[A C] = (A, C)]
119, 10eqcomi 79 . . 3 ⊤⊧[(A, C) = [A C]]
128, 7, 2a17i 106 . . . . 5 R⊧[(λx:α B) = ]
13 hbct.5 . . . . 5 R⊧[(λx:α CB) = C]
148, 4, 7, 5, 12, 1, 13hbov 111 . . . 4 R⊧[(λx:α [A C]B) = [A C]]
15 wtru 43 . . . 4 ⊤:∗
1614, 15adantr 55 . . 3 (R, ⊤)⊧[(λx:α [A C]B) = [A C]]
176, 7, 11, 16hbxfrf 107 . 2 (R, ⊤)⊧[(λx:α (A, C)B) = (A, C)]
183, 17mpdan 35 1 R⊧[(λx:α (A, C)B) = (A, C)]
 Colors of variables: type var term Syntax hints:   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12   ∧ tan 119 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-an 128 This theorem is referenced by:  alimdv  184  ax5  207
 Copyright terms: Public domain W3C validator