Theorem olcnd 878

Description: A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2024.)

Hypotheses

Ref	Expression
olcnd.1	⊢ (𝜑 → (𝜓 ∨ 𝜒))
olcnd.2	⊢ (𝜑 → ¬ 𝜒)

Assertion

Ref	Expression
olcnd	⊢ (𝜑 → 𝜓)

Proof of Theorem olcnd

Step	Hyp	Ref	Expression
1		olcnd.2	. 2 ⊢ (𝜑 → ¬ 𝜒)
2		olcnd.1	. . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
3	2	ord 865	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
4	1, 3	mt3d 148	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-or 849

This theorem is referenced by:  orcnd  879  ecase23d  1476  elprn2  4610  1sdom2dom  9159  finnzfsuppd  9281  fzone1  13705  tdeglem4  26026  ltonold  28262  xnn0nn0d  32855  ccatws1f1o  33036  mxidlirred  33557  fldextrspundgdvdslem  33850  fldext2rspun  33852  zarclssn  34043  eulerpartlemgvv  34546  lcmineqlem23  42384  chnerlem1  47203