Theorem orcom 733

Description: Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)

Assertion

Ref	Expression
orcom	⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))

Proof of Theorem orcom

Step	Hyp	Ref	Expression
1		pm1.4 732	. 2 ⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑))
2		pm1.4 732	. 2 ⊢ ((𝜓 ∨ 𝜑) → (𝜑 ∨ 𝜓))
3	1, 2	impbii 126	1 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))

Syntax hints:   ↔ wb 105   ∨ wo 713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  orcomd  734  orbi1i  768  orass  772  or32  775  or42  777  orbi1d  796  pm5.61  799  oranabs  820  ordir  822  pm2.1dc  842  notnotrdc  848  dcnnOLD  854  pm5.17dc  909  pm5.7dc  960  dn1dc  966  pm5.75  968  3orrot  1008  3orcomb  1011  excxor  1420  xorcom  1430  19.33b2  1675  nf4dc  1716  nf4r  1717  19.31r  1727  dveeq2  1861  sbequilem  1884  dvelimALT  2061  dvelimfv  2062  dvelimor  2069  eueq2dc  2976  uncom  3348  reuun2  3487  prel12  3849  exmid01  4282  exmidsssnc  4287  ordtriexmid  4613  ordtri2orexmid  4615  ontr2exmid  4617  onsucsssucexmid  4619  ordsoexmid  4654  ordtri2or2exmid  4663  cnvsom  5272  fununi  5389  frec0g  6549  frecabcl  6551  frecsuclem  6558  swoer  6716  inffiexmid  7076  exmidontriimlem1  7411  enq0tr  7629  letr  8237  reapmul1  8750  reapneg  8752  reapcotr  8753  remulext1  8754  apsym  8761  mulext1  8767  elznn0nn  9468  elznn0  9469  zapne  9529  nneoor  9557  nn01to3  9820  ltxr  9979  xrletr  10012  swrdnd  11199  maxclpr  11741  minclpr  11756  odd2np1lem  12391  lcmcom  12594  dvdsprime  12652  coprm  12674  opprdomnbg  14246  bdbl  15185  cos11  15535  lgsdir2lem4  15718  subctctexmid  16395