Theorem orim12i 767

Description: Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)

Hypotheses

Ref	Expression
orim12i.1	⊢ (𝜑 → 𝜓)
orim12i.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
orim12i	⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃))

Proof of Theorem orim12i

Step	Hyp	Ref	Expression
1		orim12i.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	orcd 741	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜃))
3		orim12i.2	. . 3 ⊢ (𝜒 → 𝜃)
4	3	olcd 742	. 2 ⊢ (𝜒 → (𝜓 ∨ 𝜃))
5	2, 4	jaoi 724	1 ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃))

Syntax hints:   → wi 4   ∨ wo 716

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 717

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  orim1i  768  orim2i  769  dcim  849  pm5.12dc  918  pm5.14dc  919  pm5.55dc  921  pm5.54dc  926  prlem2  983  ifpdc  988  ifpor  996  xordc1  1438  19.43  1677  eueq3dc  2994  inssun  3465  abvor0dc  3536  ifmdc  3669  undifexmid  4311  pwssunim  4410  ordtriexmid  4648  ontriexmidim  4649  ordtri2orexmid  4650  ontr2exmid  4652  onsucsssucexmid  4654  onsucelsucexmid  4657  ordsoexmid  4689  0elsucexmid  4692  ordpwsucexmid  4697  ordtri2or2exmid  4698  ontri2orexmidim  4699  funcnvuni  5430  oprabidlem  6089  2oconcl  6685  inffiexmid  7179  unfiexmid  7191  ctssexmid  7454  exmidonfinlem  7509  sup3exmid  9248  zeo  9701  ef0lem  12371