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Mirrors > Home > MPE Home > Th. List > Mathboxes > tsxo4 | Structured version Visualization version GIF version |
Description: A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
Ref | Expression |
---|---|
tsxo4 | ⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsbi4 36290 | . 2 ⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))) | |
2 | df-xor 1507 | . . . 4 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) | |
3 | 2 | bicomi 223 | . . 3 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ⊻ 𝜓)) |
4 | 3 | orbi2i 910 | . 2 ⊢ (((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓))) |
5 | 1, 4 | sylib 217 | 1 ⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ (𝜑 ⊻ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∨ wo 844 ⊻ wxo 1506 |
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-or 845 df-xor 1507 |
This theorem is referenced by: (None) |
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