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  4597  1sdom2dom  9164  finnzfsuppd  9286  fzone1  13739  tdeglem4  26025  ltonold  28253  xnn0nn0d  32845  ccatws1f1o  33011  mxidlirred  33532  fldextrspundgdvdslem  33824  fldext2rspun  33826  zarclssn  34017  eulerpartlemgvv  34520  lcmineqlem23  42490  chnerlem1  47310