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Mirrors > Home > ILE Home > Th. List > 3ancoma | GIF version |
Description: Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.) |
Ref | Expression |
---|---|
3ancoma | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
2 | 1 | anbi1i 451 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒)) |
3 | df-3an 945 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
4 | df-3an 945 | . 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒)) | |
5 | 2, 3, 4 | 3bitr4i 211 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 943 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 945 |
This theorem is referenced by: 3ancomb 951 3anrev 953 3anan12 955 3com12 1166 elfzmlbp 9796 elfzo2 9814 |
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