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Mirrors > Home > MPE Home > Th. List > tbwlem1 | 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 |
---|---|
tbwlem1 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tbw-ax2 1695 | . . . 4 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜓)) | |
2 | tbw-ax1 1694 | . . . 4 ⊢ (((𝜓 → 𝜒) → 𝜓) → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒))) | |
3 | 1, 2 | tbwsyl 1698 | . . 3 ⊢ (𝜓 → ((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒))) |
4 | tbw-ax1 1694 | . . . 4 ⊢ (((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((((𝜓 → 𝜒) → 𝜒) → 𝜒) → ((𝜓 → 𝜒) → 𝜒))) | |
5 | tbw-ax3 1696 | . . . 4 ⊢ (((((𝜓 → 𝜒) → 𝜒) → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((𝜓 → 𝜒) → 𝜒)) | |
6 | 4, 5 | tbwsyl 1698 | . . 3 ⊢ (((𝜓 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)) → ((𝜓 → 𝜒) → 𝜒)) |
7 | 3, 6 | tbwsyl 1698 | . 2 ⊢ (𝜓 → ((𝜓 → 𝜒) → 𝜒)) |
8 | tbw-ax1 1694 | . 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒))) | |
9 | tbw-ax1 1694 | . 2 ⊢ ((𝜓 → ((𝜓 → 𝜒) → 𝜒)) → ((((𝜓 → 𝜒) → 𝜒) → (𝜑 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))) | |
10 | 7, 8, 9 | mpsyl 68 | 1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: tbwlem2 1700 tbwlem4 1702 tbwlem5 1703 re1luk3 1706 |
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