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Theorem hbov 111
 Description: Hypothesis builder for binary operation. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
hbov.1 F:(β → (γδ))
hbov.2 A:β
hbov.3 B:α
hbov.4 C:γ
hbov.5 R⊧[(λx:α FB) = F]
hbov.6 R⊧[(λx:α AB) = A]
hbov.7 R⊧[(λx:α CB) = C]
Assertion
Ref Expression
hbov R⊧[(λx:α [AFC]B) = [AFC]]

Proof of Theorem hbov
StepHypRef Expression
1 hbov.5 . . . 4 R⊧[(λx:α FB) = F]
21ax-cb1 29 . . 3 R:∗
32trud 27 . 2 R⊧⊤
4 hbov.1 . . . 4 F:(β → (γδ))
5 hbov.2 . . . 4 A:β
6 hbov.4 . . . 4 C:γ
74, 5, 6wov 72 . . 3 [AFC]:δ
8 hbov.3 . . 3 B:α
9 weq 41 . . . 4 = :(δ → (δ → ∗))
104, 5wc 50 . . . . 5 (FA):(γδ)
1110, 6wc 50 . . . 4 ((FA)C):δ
124, 5, 6df-ov 73 . . . 4 ⊤⊧(( = [AFC])((FA)C))
139, 7, 11, 12dfov2 75 . . 3 ⊤⊧[[AFC] = ((FA)C)]
14 hbov.6 . . . . . 6 R⊧[(λx:α AB) = A]
154, 5, 8, 1, 14hbc 110 . . . . 5 R⊧[(λx:α (FA)B) = (FA)]
16 hbov.7 . . . . 5 R⊧[(λx:α CB) = C]
1710, 6, 8, 15, 16hbc 110 . . . 4 R⊧[(λx:α ((FA)C)B) = ((FA)C)]
18 wtru 43 . . . 4 ⊤:∗
1917, 18adantr 55 . . 3 (R, ⊤)⊧[(λx:α ((FA)C)B) = ((FA)C)]
207, 8, 13, 19hbxfrf 107 . 2 (R, ⊤)⊧[(λx:α [AFC]B) = [AFC]]
213, 20mpdan 35 1 R⊧[(λx:α [AFC]B) = [AFC]]
 Colors of variables: type var term Syntax hints:   → ht 2  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12 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-wl 65  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  clf  115  hbct  155  exlimdv  167  cbvf  179  leqf  181  exlimd  183  exmid  199  axrep  220
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