Theorem hbov 111

Description: Hypothesis builder for binary operation. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
hbov.1	⊢ F:(β → (γ → δ))
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

Step	Hyp	Ref	Expression
1		hbov.5	. . . 4 ⊢ R⊧[(λx:α FB) = F]
2	1	ax-cb1 29	. . 3 ⊢ R:∗
3	2	trud 27	. 2 ⊢ R⊧⊤
4		hbov.1	. . . 4 ⊢ F:(β → (γ → δ))
5		hbov.2	. . . 4 ⊢ A:β
6		hbov.4	. . . 4 ⊢ C:γ
7	4, 5, 6	wov 72	. . 3 ⊢ [AFC]:δ
8		hbov.3	. . 3 ⊢ B:α
9		weq 41	. . . 4 ⊢ = :(δ → (δ → ∗))
10	4, 5	wc 50	. . . . 5 ⊢ (FA):(γ → δ)
11	10, 6	wc 50	. . . 4 ⊢ ((FA)C):δ
12	4, 5, 6	df-ov 73	. . . 4 ⊢ ⊤⊧(( = [AFC])((FA)C))
13	9, 7, 11, 12	dfov2 75	. . 3 ⊢ ⊤⊧[[AFC] = ((FA)C)]
14		hbov.6	. . . . . 6 ⊢ R⊧[(λx:α AB) = A]
15	4, 5, 8, 1, 14	hbc 110	. . . . 5 ⊢ R⊧[(λx:α (FA)B) = (FA)]
16		hbov.7	. . . . 5 ⊢ R⊧[(λx:α CB) = C]
17	10, 6, 8, 15, 16	hbc 110	. . . 4 ⊢ R⊧[(λx:α ((FA)C)B) = ((FA)C)]
18		wtru 43	. . . 4 ⊢ ⊤:∗
19	17, 18	adantr 55	. . 3 ⊢ (R, ⊤)⊧[(λx:α ((FA)C)B) = ((FA)C)]
20	7, 8, 13, 19	hbxfrf 107	. 2 ⊢ (R, ⊤)⊧[(λx:α [AFC]B) = [AFC]]
21	3, 20	mpdan 35	1 ⊢ R⊧[(λx:α [AFC]B) = [AFC]]

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