Theorem an32 562

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 401	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
2		an12 561	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒)))
3		ancom 266	. 2 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))
4	1, 2, 3	3bitri 206	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  an32s  568  3anan32  989  indifdir  3392  inrab2  3409  reupick  3420  unidif0  4168  resco  5134  f11o  5495  respreima  5645  dff1o6  5777  dfoprab2  5922  xpassen  6830  enq0enq  7430  elioomnf  9968  modfsummod  11466  pcqcl  12306  tx1cn  13772  isms2  13957  elcncf1di  14069