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Mirrors > Home > ILE Home > Th. List > 3orbi123d | GIF version |
Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
Ref | Expression |
---|---|
bi3d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bi3d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
bi3d.3 | ⊢ (𝜑 → (𝜂 ↔ 𝜁)) |
Ref | Expression |
---|---|
3orbi123d | ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi3d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | bi3d.2 | . . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
3 | 1, 2 | orbi12d 745 | . . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) |
4 | bi3d.3 | . . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁)) | |
5 | 3, 4 | orbi12d 745 | . 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁))) |
6 | df-3or 928 | . 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) | |
7 | df-3or 928 | . 2 ⊢ ((𝜒 ∨ 𝜏 ∨ 𝜁) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁)) | |
8 | 5, 6, 7 | 3bitr4g 222 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 667 ∨ w3o 926 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 |
This theorem depends on definitions: df-bi 116 df-3or 928 |
This theorem is referenced by: ordtriexmid 4366 wetriext 4420 nntri3or 6294 tridc 6695 ltsopi 6976 pitri3or 6978 nqtri3or 7052 elz 8850 ztri3or 8891 qtri3or 9803 |
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