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Mirrors > Home > HOLE Home > Th. List > hbxfrf | GIF version |
Description: Transfer a hypothesis builder to an equivalent expression. |
Ref | Expression |
---|---|
hbxfr.1 | ⊢ T:β |
hbxfr.2 | ⊢ B:α |
hbxfrf.3 | ⊢ R⊧[T = A] |
hbxfrf.4 | ⊢ (S, R)⊧[(λx:α AB) = A] |
Ref | Expression |
---|---|
hbxfrf | ⊢ (S, R)⊧[(λx:α TB) = T] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbxfr.1 | . . . . 5 ⊢ T:β | |
2 | hbxfrf.3 | . . . . 5 ⊢ R⊧[T = A] | |
3 | 1, 2 | eqtypi 69 | . . . 4 ⊢ A:β |
4 | 3 | wl 59 | . . 3 ⊢ λx:α A:(α → β) |
5 | hbxfr.2 | . . 3 ⊢ B:α | |
6 | 4, 5 | wc 45 | . 2 ⊢ (λx:α AB):β |
7 | hbxfrf.4 | . 2 ⊢ (S, R)⊧[(λx:α AB) = A] | |
8 | 1 | wl 59 | . . . 4 ⊢ λx:α T:(α → β) |
9 | 1, 2 | leq 81 | . . . 4 ⊢ R⊧[λx:α T = λx:α A] |
10 | 8, 5, 9 | ceq1 79 | . . 3 ⊢ R⊧[(λx:α TB) = (λx:α AB)] |
11 | 7 | ax-cb1 29 | . . . 4 ⊢ (S, R):∗ |
12 | 11 | wctl 31 | . . 3 ⊢ S:∗ |
13 | 10, 12 | adantl 51 | . 2 ⊢ (S, R)⊧[(λx:α TB) = (λx:α AB)] |
14 | 2, 12 | adantl 51 | . 2 ⊢ (S, R)⊧[T = A] |
15 | 6, 7, 13, 14 | 3eqtr4i 86 | 1 ⊢ (S, R)⊧[(λx:α TB) = T] |
Colors of variables: type var term |
Syntax hints: kc 5 λkl 6 = ke 7 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 |
This theorem was proved from axioms: ax-syl 15 ax-jca 17 ax-simpl 20 ax-simpr 21 ax-trud 26 ax-cb1 29 ax-cb2 30 ax-refl 39 ax-eqmp 42 ax-ceq 46 ax-leq 62 |
This theorem depends on definitions: df-ov 65 |
This theorem is referenced by: hbxfr 98 hbov 101 hbct 145 |
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