Theorem orim12i 764

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 738	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜃))
3		orim12i.2	. . 3 ⊢ (𝜒 → 𝜃)
4	3	olcd 739	. 2 ⊢ (𝜒 → (𝜓 ∨ 𝜃))
5	2, 4	jaoi 721	1 ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃))

Syntax hints:   → wi 4   ∨ 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:  orim1i  765  orim2i  766  dcim  846  pm5.12dc  915  pm5.14dc  916  pm5.55dc  918  pm5.54dc  923  prlem2  980  ifpdc  985  ifpor  993  xordc1  1435  19.43  1674  eueq3dc  2977  inssun  3444  abvor0dc  3515  ifmdc  3645  undifexmid  4277  pwssunim  4375  ordtriexmid  4613  ontriexmidim  4614  ordtri2orexmid  4615  ontr2exmid  4617  onsucsssucexmid  4619  onsucelsucexmid  4622  ordsoexmid  4654  0elsucexmid  4657  ordpwsucexmid  4662  ordtri2or2exmid  4663  ontri2orexmidim  4664  funcnvuni  5390  oprabidlem  6038  2oconcl  6593  inffiexmid  7076  unfiexmid  7088  ctssexmid  7325  exmidonfinlem  7379  sup3exmid  9112  zeo  9560  ef0lem  12179