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Mirrors > Home > HOLE Home > Th. List > alval | GIF version |
Description: Value of the for all predicate. |
Ref | Expression |
---|---|
alval.1 | ⊢ F:(α → ∗) |
Ref | Expression |
---|---|
alval | ⊢ ⊤⊧[(∀F) = [F = λx:α ⊤]] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wal 124 | . . 3 ⊢ ∀:((α → ∗) → ∗) | |
2 | alval.1 | . . 3 ⊢ F:(α → ∗) | |
3 | 1, 2 | wc 45 | . 2 ⊢ (∀F):∗ |
4 | df-al 116 | . . 3 ⊢ ⊤⊧[∀ = λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]] | |
5 | 1, 2, 4 | ceq1 79 | . 2 ⊢ ⊤⊧[(∀F) = (λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]F)] |
6 | wv 58 | . . . 4 ⊢ p:(α → ∗):(α → ∗) | |
7 | wtru 40 | . . . . 5 ⊢ ⊤:∗ | |
8 | 7 | wl 59 | . . . 4 ⊢ λx:α ⊤:(α → ∗) |
9 | 6, 8 | weqi 68 | . . 3 ⊢ [p:(α → ∗) = λx:α ⊤]:∗ |
10 | weq 38 | . . . 4 ⊢ = :((α → ∗) → ((α → ∗) → ∗)) | |
11 | 6, 2 | weqi 68 | . . . . 5 ⊢ [p:(α → ∗) = F]:∗ |
12 | 11 | id 25 | . . . 4 ⊢ [p:(α → ∗) = F]⊧[p:(α → ∗) = F] |
13 | 10, 6, 8, 12 | oveq1 89 | . . 3 ⊢ [p:(α → ∗) = F]⊧[[p:(α → ∗) = λx:α ⊤] = [F = λx:α ⊤]] |
14 | 9, 2, 13 | cl 106 | . 2 ⊢ ⊤⊧[(λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]F) = [F = λx:α ⊤]] |
15 | 3, 5, 14 | eqtri 85 | 1 ⊢ ⊤⊧[(∀F) = [F = λx:α ⊤]] |
Colors of variables: type var term |
Syntax hints: tv 1 → ht 2 ∗hb 3 kc 5 λkl 6 = ke 7 ⊤kt 8 [kbr 9 ⊧wffMMJ2 11 wffMMJ2t 12 ∀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-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-hbl1 93 ax-17 95 ax-inst 103 |
This theorem depends on definitions: df-ov 65 df-al 116 |
This theorem is referenced by: ax4g 139 alrimiv 141 olc 154 orc 155 alrimi 170 |
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