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  8125  mndifsplit  22861  maducoeval2  22865  leibpilem2  27181  leibpi  27182  3o1cs  32941  3o2cs  32942  elrgspnlem2  33686  poimirlem31  38403  tsan2  38893  frege114d  44601  ntrneiel2  44929  nnfoctbdjlem  47286  homf0  49938