Theorem an13 774

Description: A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)

Assertion

Ref	Expression
an13	⊢ ((φ ∧ (ψ ∧ χ)) ↔ (χ ∧ (ψ ∧ φ)))

Proof of Theorem an13

Step	Hyp	Ref	Expression
1		an12 772	. 2 ⊢ ((φ ∧ (ψ ∧ χ)) ↔ (ψ ∧ (φ ∧ χ)))
2		anass 630	. 2 ⊢ (((ψ ∧ φ) ∧ χ) ↔ (ψ ∧ (φ ∧ χ)))
3		ancom 437	. 2 ⊢ (((ψ ∧ φ) ∧ χ) ↔ (χ ∧ (ψ ∧ φ)))
4	1, 2, 3	3bitr2i 264	1 ⊢ ((φ ∧ (ψ ∧ χ)) ↔ (χ ∧ (ψ ∧ φ)))

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

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

This theorem is used by:  an31  775  eqvinop  4607  elxp2  4803  iunfopab  5205