Theorem 3ancoma 588

Description: Commutation law for triple conjunction.

Assertion

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

Proof of Theorem 3ancoma

Step	Hyp	Ref	Expression
1		ancom 333	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
2	1	anbi1i 368	. 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((ψ ∧ φ) ∧ χ))
3		df-3an 583	. 2 ⊢ ((φ ∧ ψ ∧ χ) ↔ ((φ ∧ ψ) ∧ χ))
4		df-3an 583	. 2 ⊢ ((ψ ∧ φ ∧ χ) ↔ ((ψ ∧ φ) ∧ χ))
5	2, 3, 4	3bitr4 158	1 ⊢ ((φ ∧ ψ ∧ χ) ↔ (ψ ∧ φ ∧ χ))

Syntax hints:   ↔ wb 127   ∧ wa 196   ∧ w3a 581

This theorem is referenced by:  3ancomb 589  3anrev 590

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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