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Mirrors > Home > MPE Home > Th. List > Mathboxes > tsna1 | Structured version Visualization version GIF version |
Description: A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
Ref | Expression |
---|---|
tsna1 | ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsan1 36066 | . 2 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))) | |
2 | notnotb 318 | . . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ ¬ (𝜑 ⊼ 𝜓)) | |
3 | df-nan 1488 | . . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitr3i 280 | . . . 4 ⊢ (¬ ¬ (𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) |
5 | 4 | con4bii 324 | . . 3 ⊢ (¬ (𝜑 ⊼ 𝜓) ↔ (𝜑 ∧ 𝜓)) |
6 | 5 | orbi2i 913 | . 2 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))) |
7 | 1, 6 | sylibr 237 | 1 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 847 ⊼ wnan 1487 |
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 210 df-an 400 df-or 848 df-nan 1488 |
This theorem is referenced by: (None) |
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