Theorem orcs 889

Description: Deduction eliminating disjunct. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 18) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)

Hypothesis

Ref	Expression
orcs.1	⊢ ((𝜑 ∨ 𝜓) → 𝜒)

Assertion

Ref	Expression
orcs	⊢ (𝜑 → 𝜒)

Proof of Theorem orcs

Step	Hyp	Ref	Expression
1		orc 881	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2		orcs.1	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜒)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by:  olcs  890  norasslem2  1565  ifor  4537  tppreqb  4768  frxp  8127  mndifsplit  22931  maducoeval2  22935  leibpilem2  27251  leibpi  27252  3o1cs  33041  3o2cs  33042  elrgspnlem2  33786  poimirlem31  38537  tsan2  39042  frege114d  44717  ntrneiel2  45045  nnfoctbdjlem  47409  homf0  50061