Theorem orcs 881

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 873	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2		orcs.1	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜒)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∨ wo 853

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 208  df-or 854

This theorem is referenced by:  olcs  882  norasslem2  1542  ifor  4516  tppreqb  4745  frxp  8073  mndifsplit  22626  maducoeval2  22630  leibpilem2  26930  leibpi  26931  3o1cs  32555  3o2cs  32556  elrgspnlem2  33331  poimirlem31  38025  tsan2  38516  frege114d  44209  ntrneiel2  44537  nnfoctbdjlem  46905  homf0  49506