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Mirrors > Home > MPE Home > Th. List > Mathboxes > tsbi1 | Structured version Visualization version GIF version |
Description: A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
Ref | Expression |
---|---|
tsbi1 | ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.1 898 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓)) | |
2 | 1 | olcd 407 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) |
3 | pm3.13 521 | . . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)) | |
4 | 3 | orcd 406 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) |
5 | 2, 4 | pm2.61i 175 | . 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓)) |
6 | 5 | a1i 11 | 1 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 |
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 196 df-or 384 df-an 385 |
This theorem is referenced by: tsxo1 33114 mpt2bi123f 33141 mptbi12f 33145 ac6s6 33150 |
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