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  991  indifdir  3420  inrab2  3437  reupick  3448  unidif0  4201  resco  5175  f11o  5540  respreima  5693  dff1o6  5826  dfoprab2  5973  xpassen  6898  enq0enq  7515  elioomnf  10060  modfsummod  11640  pcqcl  12500  tx1cn  14589  isms2  14774  elcncf1di  14899