Higher-Order Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HOLE Home > Th. List > con2d | GIF version |
Description: A contraposition deduction. |
Ref | Expression |
---|---|
con2d.1 | ⊢ T:∗ |
con2d.2 | ⊢ (R, S)⊧(¬ T) |
Ref | Expression |
---|---|
con2d | ⊢ (R, T)⊧(¬ S) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con2d.1 | . . . . 5 ⊢ T:∗ | |
2 | wfal 125 | . . . . 5 ⊢ ⊥:∗ | |
3 | con2d.2 | . . . . . 6 ⊢ (R, S)⊧(¬ T) | |
4 | 3 | ax-cb1 29 | . . . . . . 7 ⊢ (R, S):∗ |
5 | 1 | notval 135 | . . . . . . 7 ⊢ ⊤⊧[(¬ T) = [T ⇒ ⊥]] |
6 | 4, 5 | a1i 28 | . . . . . 6 ⊢ (R, S)⊧[(¬ T) = [T ⇒ ⊥]] |
7 | 3, 6 | mpbi 72 | . . . . 5 ⊢ (R, S)⊧[T ⇒ ⊥] |
8 | 1, 2, 7 | imp 147 | . . . 4 ⊢ ((R, S), T)⊧⊥ |
9 | 8 | an32s 55 | . . 3 ⊢ ((R, T), S)⊧⊥ |
10 | 9 | ex 148 | . 2 ⊢ (R, T)⊧[S ⇒ ⊥] |
11 | 4 | wctl 31 | . . . 4 ⊢ R:∗ |
12 | 11, 1 | wct 44 | . . 3 ⊢ (R, T):∗ |
13 | 4 | wctr 32 | . . . 4 ⊢ S:∗ |
14 | 13 | notval 135 | . . 3 ⊢ ⊤⊧[(¬ S) = [S ⇒ ⊥]] |
15 | 12, 14 | a1i 28 | . 2 ⊢ (R, T)⊧[(¬ S) = [S ⇒ ⊥]] |
16 | 10, 15 | mpbir 77 | 1 ⊢ (R, T)⊧(¬ S) |
Colors of variables: type var term |
Syntax hints: ∗hb 3 kc 5 = ke 7 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ⊥tfal 108 ¬ tne 110 ⇒ tim 111 |
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-al 116 df-fal 117 df-an 118 df-im 119 df-not 120 |
This theorem is referenced by: con3d 152 exnal1 175 |
Copyright terms: Public domain | W3C validator |