Theorem an32 496

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 381	. 2 ⊢ (((φ ∧ ψ) ∧ χ) ↔ (φ ∧ (ψ ∧ χ)))
2		an12 495	. 2 ⊢ ((φ ∧ (ψ ∧ χ)) ↔ (ψ ∧ (φ ∧ χ)))
3		ancom 253	. 2 ⊢ ((ψ ∧ (φ ∧ χ)) ↔ ((φ ∧ χ) ∧ ψ))
4	1, 2, 3	3bitri 195	1 ⊢ (((φ ∧ ψ) ∧ χ) ↔ ((φ ∧ χ) ∧ ψ))

Syntax hints:   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  an32s  502  3anan32  895  indifdir  3187  inrab2  3204  reupick  3215  unidif0  3911  resco  4768  f11o  5102  respreima  5238  dff1o6  5359  dfoprab2  5494  xpassen  6240  enq0enq  6414  elioomnf  8607