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Mirrors > Home > MPE Home > Th. List > tbwlem2 | Structured version Visualization version GIF version |
Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
tbwlem2 | ⊢ ((𝜑 → (𝜓 → ⊥)) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tbw-ax4 1705 | . . . . 5 ⊢ (⊥ → 𝜒) | |
2 | tbw-ax1 1702 | . . . . . 6 ⊢ ((𝜓 → ⊥) → ((⊥ → 𝜒) → (𝜓 → 𝜒))) | |
3 | tbwlem1 1707 | . . . . . 6 ⊢ (((𝜓 → ⊥) → ((⊥ → 𝜒) → (𝜓 → 𝜒))) → ((⊥ → 𝜒) → ((𝜓 → ⊥) → (𝜓 → 𝜒)))) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ ((⊥ → 𝜒) → ((𝜓 → ⊥) → (𝜓 → 𝜒))) |
5 | 1, 4 | ax-mp 5 | . . . 4 ⊢ ((𝜓 → ⊥) → (𝜓 → 𝜒)) |
6 | tbwlem1 1707 | . . . 4 ⊢ (((𝜓 → ⊥) → (𝜓 → 𝜒)) → (𝜓 → ((𝜓 → ⊥) → 𝜒))) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ (𝜓 → ((𝜓 → ⊥) → 𝜒)) |
8 | tbw-ax1 1702 | . . 3 ⊢ ((𝜑 → (𝜓 → ⊥)) → (((𝜓 → ⊥) → 𝜒) → (𝜑 → 𝜒))) | |
9 | tbw-ax1 1702 | . . 3 ⊢ ((𝜓 → ((𝜓 → ⊥) → 𝜒)) → ((((𝜓 → ⊥) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))) | |
10 | 7, 8, 9 | mpsyl 68 | . 2 ⊢ ((𝜑 → (𝜓 → ⊥)) → (𝜓 → (𝜑 → 𝜒))) |
11 | tbw-ax1 1702 | . 2 ⊢ ((𝜓 → (𝜑 → 𝜒)) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃))) | |
12 | 10, 11 | tbwsyl 1706 | 1 ⊢ ((𝜑 → (𝜓 → ⊥)) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊥wfal 1553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-tru 1544 df-fal 1554 |
This theorem is referenced by: tbwlem4 1710 |
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