![]() |
Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
|
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 | pm5.1 812 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓)) | |
2 | 1 | olcd 860 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) |
3 | pm3.13 977 | . . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)) | |
4 | 3 | orcd 859 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ↔ 𝜓))) |
5 | 2, 4 | pm2.61i 177 | . 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 198 ∧ wa 387 ∨ wo 833 |
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 199 df-an 388 df-or 834 |
This theorem is referenced by: tsxo1 34856 mpobi123f 34881 mptbi12f 34885 ac6s6 34891 |
Copyright terms: Public domain | W3C validator |