Theorem 3ancoma 985

Description: Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Assertion

Ref	Expression
3ancoma	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒))

Proof of Theorem 3ancoma

Step	Hyp	Ref	Expression
1		ancom 266	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
2	1	anbi1i 458	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒))
3		df-3an 980	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
4		df-3an 980	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒))
5	2, 3, 4	3bitr4i 212	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 980

This theorem is referenced by:  3ancomb  986  3anrev  988  3anan12  990  3com12  1207  elfzmlbp  10134  elfzo2  10152  pythagtriplem2  12268  pythagtrip  12285  xpsfrnel  12768