Theorem orcs 874

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 866	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2		orcs.1	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜒)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∨ wo 846

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 847

This theorem is referenced by:  olcs  875  ifor  4602  tppreqb  4830  frxp  8167  mndifsplit  22663  maducoeval2  22667  leibpilem2  27002  leibpi  27003  3o1cs  32490  3o2cs  32491  poimirlem31  37611  tsan2  38102  frege114d  43720  ntrneiel2  44048  nnfoctbdjlem  46376