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Mirrors > Home > HOLE Home > Th. List > hbct | GIF version |
Description: Hypothesis builder for context conjunction. |
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] |
Ref | Expression |
---|---|
hbct | ⊢ R⊧[(λx:α (A, C)B) = (A, C)] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbct.4 | . . . 4 ⊢ R⊧[(λx:α AB) = A] | |
2 | 1 | ax-cb1 29 | . . 3 ⊢ R:∗ |
3 | 2 | trud 27 | . 2 ⊢ R⊧⊤ |
4 | hbct.1 | . . . 4 ⊢ A:∗ | |
5 | hbct.3 | . . . 4 ⊢ C:∗ | |
6 | 4, 5 | wct 44 | . . 3 ⊢ (A, C):∗ |
7 | hbct.2 | . . 3 ⊢ B:α | |
8 | wan 126 | . . . . 5 ⊢ ∧ :(∗ → (∗ → ∗)) | |
9 | 8, 4, 5 | wov 64 | . . . 4 ⊢ [A ∧ C]:∗ |
10 | 4, 5 | dfan2 144 | . . . 4 ⊢ ⊤⊧[[A ∧ C] = (A, C)] |
11 | 9, 10 | eqcomi 70 | . . 3 ⊢ ⊤⊧[(A, C) = [A ∧ C]] |
12 | 8, 7, 2 | a17i 96 | . . . . 5 ⊢ R⊧[(λx:α ∧ B) = ∧ ] |
13 | hbct.5 | . . . . 5 ⊢ R⊧[(λx:α CB) = C] | |
14 | 8, 4, 7, 5, 12, 1, 13 | hbov 101 | . . . 4 ⊢ R⊧[(λx:α [A ∧ C]B) = [A ∧ C]] |
15 | wtru 40 | . . . 4 ⊢ ⊤:∗ | |
16 | 14, 15 | adantr 50 | . . 3 ⊢ (R, ⊤)⊧[(λx:α [A ∧ C]B) = [A ∧ C]] |
17 | 6, 7, 11, 16 | hbxfrf 97 | . 2 ⊢ (R, ⊤)⊧[(λx:α (A, C)B) = (A, C)] |
18 | 3, 17 | mpdan 33 | 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 109 |
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-ded 43 ax-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-distrl 63 ax-hbl1 93 ax-17 95 ax-inst 103 |
This theorem depends on definitions: df-ov 65 df-an 118 |
This theorem is referenced by: alimdv 172 ax5 194 |
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