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Mirrors > Home > HOLE Home > Th. List > hbov | GIF version |
Description: Hypothesis builder for binary operation. |
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] |
Ref | Expression |
---|---|
hbov | ⊢ R⊧[(λx:α [AFC]B) = [AFC]] |
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 64 | . . 3 ⊢ [AFC]:δ |
8 | hbov.3 | . . 3 ⊢ B:α | |
9 | weq 38 | . . . 4 ⊢ = :(δ → (δ → ∗)) | |
10 | 4, 5 | wc 45 | . . . . 5 ⊢ (FA):(γ → δ) |
11 | 10, 6 | wc 45 | . . . 4 ⊢ ((FA)C):δ |
12 | 4, 5, 6 | df-ov 65 | . . . 4 ⊢ ⊤⊧(( = [AFC])((FA)C)) |
13 | 9, 7, 11, 12 | dfov2 67 | . . 3 ⊢ ⊤⊧[[AFC] = ((FA)C)] |
14 | hbov.6 | . . . . . 6 ⊢ R⊧[(λx:α AB) = A] | |
15 | 4, 5, 8, 1, 14 | hbc 100 | . . . . 5 ⊢ R⊧[(λx:α (FA)B) = (FA)] |
16 | hbov.7 | . . . . 5 ⊢ R⊧[(λx:α CB) = C] | |
17 | 10, 6, 8, 15, 16 | hbc 100 | . . . 4 ⊢ R⊧[(λx:α ((FA)C)B) = ((FA)C)] |
18 | wtru 40 | . . . 4 ⊢ ⊤:∗ | |
19 | 17, 18 | adantr 50 | . . 3 ⊢ (R, ⊤)⊧[(λx:α ((FA)C)B) = ((FA)C)] |
20 | 7, 8, 13, 19 | hbxfrf 97 | . 2 ⊢ (R, ⊤)⊧[(λx:α [AFC]B) = [AFC]] |
21 | 3, 20 | mpdan 33 | 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-refl 39 ax-eqmp 42 ax-ceq 46 ax-distrc 61 ax-leq 62 |
This theorem depends on definitions: df-ov 65 |
This theorem is referenced by: clf 105 hbct 145 exlimdv 157 cbvf 167 leqf 169 exlimd 171 exmid 186 axrep 207 |
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