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Mirrors > Home > ILE Home > Th. List > orbi12d | GIF version |
Description: Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
orbi12d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
orbi12d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Ref | Expression |
---|---|
orbi12d | ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orbi12d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | orbi1d 786 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜃))) |
3 | orbi12d.2 | . . 3 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
4 | 3 | orbi2d 785 | . 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) |
5 | 2, 4 | bitrd 187 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) |
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