Theorem orcs 873

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 17) -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 865	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2		orcs.1	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜒)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∨ wo 845

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 206  df-or 846

This theorem is referenced by:  olcs  874  ifor  4545  tppreqb  4770  frxp  8063  mndifsplit  22022  maducoeval2  22026  leibpilem2  26328  leibpi  26329  3o1cs  31455  3o2cs  31456  poimirlem31  36182  tsan2  36674  frege114d  42152  ntrneiel2  42480  nnfoctbdjlem  44816