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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 793 | . . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) |
4 | bi3d.3 | . . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁)) | |
5 | 3, 4 | orbi12d 793 | . 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁))) |
6 | df-3or 979 | . 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) | |
7 | df-3or 979 | . 2 ⊢ ((𝜒 ∨ 𝜏 ∨ 𝜁) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁)) | |
8 | 5, 6, 7 | 3bitr4g 223 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 708 ∨ w3o 977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-3or 979 |
This theorem is referenced by: ordtriexmid 4520 ontriexmidim 4521 wetriext 4576 nntri3or 6493 tridc 6898 exmidontriimlem3 7221 exmidontriimlem4 7222 exmidontriim 7223 onntri35 7235 ltsopi 7318 pitri3or 7320 nqtri3or 7394 elz 9254 ztri3or 9295 qtri3or 10242 trilpo 14761 trirec0 14762 reap0 14776 |
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