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Mirrors > Home > ILE Home > Th. List > or42 | GIF version |
Description: Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.) |
Ref | Expression |
---|---|
or42 | ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | or4 688 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃))) | |
2 | orcom 647 | . . 3 ⊢ ((𝜓 ∨ 𝜃) ↔ (𝜃 ∨ 𝜓)) | |
3 | 2 | orbi2i 679 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓))) |
4 | 1, 3 | bitri 173 | 1 ⊢ (((𝜑 ∨ 𝜓) ∨ (𝜒 ∨ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∨ (𝜃 ∨ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ wo 629 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: reapcotr 7589 |
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