Theorem orcs 888

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 880	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2		orcs.1	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜒)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∨ wo 860

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 210  df-or 861

This theorem is referenced by:  olcs  889  norasslem2  1558  ifor  4538  tppreqb  4768  frxp  8110  mndifsplit  22754  maducoeval2  22758  leibpilem2  27064  leibpi  27065  3o1cs  32718  3o2cs  32719  elrgspnlem2  33476  poimirlem31  38162  tsan2  38653  frege114d  44346  ntrneiel2  44674  nnfoctbdjlem  47027  homf0  49638