Higher-Order Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HOLE Home > Th. List > ax4g | GIF version |
Description: If F is true for all x:α, then it is true for A. |
Ref | Expression |
---|---|
ax4g.1 | ⊢ F:(α → ∗) |
ax4g.2 | ⊢ A:α |
Ref | Expression |
---|---|
ax4g | ⊢ (∀F)⊧(FA) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wal 124 | . . . 4 ⊢ ∀:((α → ∗) → ∗) | |
2 | ax4g.1 | . . . 4 ⊢ F:(α → ∗) | |
3 | 1, 2 | wc 45 | . . 3 ⊢ (∀F):∗ |
4 | 3 | trud 27 | . 2 ⊢ (∀F)⊧⊤ |
5 | ax4g.2 | . . . 4 ⊢ A:α | |
6 | 2, 5 | wc 45 | . . 3 ⊢ (FA):∗ |
7 | 4 | ax-cb1 29 | . . . . . 6 ⊢ (∀F):∗ |
8 | 7 | id 25 | . . . . 5 ⊢ (∀F)⊧(∀F) |
9 | 2 | alval 132 | . . . . . 6 ⊢ ⊤⊧[(∀F) = [F = λp:α ⊤]] |
10 | 7, 9 | a1i 28 | . . . . 5 ⊢ (∀F)⊧[(∀F) = [F = λp:α ⊤]] |
11 | 8, 10 | mpbi 72 | . . . 4 ⊢ (∀F)⊧[F = λp:α ⊤] |
12 | 2, 5, 11 | ceq1 79 | . . 3 ⊢ (∀F)⊧[(FA) = (λp:α ⊤A)] |
13 | 5, 4 | hbth 99 | . . 3 ⊢ (∀F)⊧[(λp:α ⊤A) = ⊤] |
14 | 6, 12, 13 | eqtri 85 | . 2 ⊢ (∀F)⊧[(FA) = ⊤] |
15 | 4, 14 | mpbir 77 | 1 ⊢ (∀F)⊧(FA) |
Colors of variables: type var term |
Syntax hints: → 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-ded 43 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: ax4 140 cla4v 142 |
Copyright terms: Public domain | W3C validator |