Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 262 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
2 | 1 | anbi1i 445 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒)) |
3 | df-3an 921 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
4 | df-3an 921 | . 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒)) | |
5 | 2, 3, 4 | 3bitr4i 210 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∧ w3a 919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 df-3an 921 |
This theorem is referenced by: 3ancomb 927 3anrev 929 3anan12 931 3com12 1142 elfzmlbp 9143 elfzo2 9160 |
Copyright terms: Public domain | W3C validator |