Theorem olcnd 877

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 864	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
4	1, 3	mt3d 148	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847

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 848

This theorem is referenced by:  orcnd  878  ecase23d  1475  elprn2  4606  1sdom2dom  9148  finnzfsuppd  9267  fzone1  13694  tdeglem4  26002  sltonold  28208  xnn0nn0d  32766  ccatws1f1o  32943  mxidlirred  33448  fldextrspundgdvdslem  33704  fldext2rspun  33706  zarclssn  33897  eulerpartlemgvv  34400  lcmineqlem23  42154  chnerlem1  46994