Theorem 3ancoma 941

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 437	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
2	1	anbi1i 676	. 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((ψ ∧ φ) ∧ χ))
3		df-3an 936	. 2 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
4		df-3an 936	. 2 ⊢ ((ψ ∧ φ ∧ χ) ↔ ((ψ ∧ φ) ∧ χ))
5	2, 3, 4	3bitr4i 268	1 ⊢ ((φ ∧ ψ ∧ χ) ↔ (ψ ∧ φ ∧ χ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by:  3ancomb  943  3anrev  945  3anan12  947  3com12  1155  cnvsi  5519  oqelins4  5795  brpprod  5840