Theorem an32 773

Description: A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)

Assertion

Ref	Expression
an32	⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ ψ))

Proof of Theorem an32

Step	Hyp	Ref	Expression
1		anass 630	. 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ (φ ∧ (ψ ∧ χ)))
2		an12 772	. 2 ⊢ ((φ ∧ (ψ ∧ χ)) ↔ (ψ ∧ (φ ∧ χ)))
3		ancom 437	. 2 ⊢ ((ψ ∧ (φ ∧ χ)) ↔ ((φ ∧ χ) ∧ ψ))
4	1, 2, 3	3bitri 262	1 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ ψ))

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:  an32s  779  3anan32  946  indifdir  3512  inrab2  3529  reupick  3540  resco  5086  imadif  5172  dff1o3  5293  f11o  5316  respreima  5411  dff1o6  5476  brpprod  5840