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Mirrors > Home > HOLE Home > Th. List > alimdv | GIF version |
Description: Deduction from Theorem 19.20 of [Margaris] p. 90. |
Ref | Expression |
---|---|
alimdv.1 | ⊢ (R, A)⊧B |
Ref | Expression |
---|---|
alimdv | ⊢ (R, (∀λx:α A))⊧(∀λx:α B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alimdv.1 | . . . . . . 7 ⊢ (R, A)⊧B | |
2 | 1 | ax-cb1 29 | . . . . . 6 ⊢ (R, A):∗ |
3 | 2 | wctr 32 | . . . . 5 ⊢ A:∗ |
4 | 3 | ax4 140 | . . . 4 ⊢ (∀λx:α A)⊧A |
5 | 2 | wctl 31 | . . . 4 ⊢ R:∗ |
6 | 4, 5 | adantl 51 | . . 3 ⊢ (R, (∀λx:α A))⊧A |
7 | 6, 1 | syldan 34 | . 2 ⊢ (R, (∀λx:α A))⊧B |
8 | wv 58 | . . 3 ⊢ y:α:α | |
9 | wal 124 | . . . 4 ⊢ ∀:((α → ∗) → ∗) | |
10 | 3 | wl 59 | . . . 4 ⊢ λx:α A:(α → ∗) |
11 | 9, 10 | wc 45 | . . 3 ⊢ (∀λx:α A):∗ |
12 | 5, 8 | ax-17 95 | . . 3 ⊢ ⊤⊧[(λx:α Ry:α) = R] |
13 | 9, 8 | ax-17 95 | . . . 4 ⊢ ⊤⊧[(λx:α ∀y:α) = ∀] |
14 | 3, 8 | ax-hbl1 93 | . . . 4 ⊢ ⊤⊧[(λx:α λx:α Ay:α) = λx:α A] |
15 | 9, 10, 8, 13, 14 | hbc 100 | . . 3 ⊢ ⊤⊧[(λx:α (∀λx:α A)y:α) = (∀λx:α A)] |
16 | 5, 8, 11, 12, 15 | hbct 145 | . 2 ⊢ ⊤⊧[(λx:α (R, (∀λx:α A))y:α) = (R, (∀λx:α A))] |
17 | 7, 16 | alrimi 170 | 1 ⊢ (R, (∀λx:α A))⊧(∀λx:α B) |
Colors of variables: type var term |
Syntax hints: tv 1 → ht 2 ∗hb 3 kc 5 λkl 6 ⊤kt 8 kct 10 ⊧wffMMJ2 11 ∀tal 112 |
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 ax-eta 165 |
This theorem depends on definitions: df-ov 65 df-al 116 df-an 118 |
This theorem is referenced by: exnal1 175 |
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