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Mirrors > Home > ILE Home > Th. List > 3anrot | GIF version |
Description: Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.) |
Ref | Expression |
---|---|
3anrot | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜑)) | |
2 | 3anass 971 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒))) | |
3 | df-3an 969 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜑) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜑)) | |
4 | 1, 2, 3 | 3bitr4i 211 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 969 |
This theorem is referenced by: 3ancomb 975 3anrev 977 3simpc 985 caovlem2d 6025 nnmcan 6478 modmulconst 11749 xmetpsmet 12910 comet 13040 |
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