Theorem orcs 408

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 399	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2		orcs.1	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜒)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∨ wo 382

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 196  df-or 384

This theorem is referenced by:  olcs  409  ifor  4085  tppreqb  4277  frxp  7174  mndifsplit  20261  maducoeval2  20265  leibpilem2  24468  leibpi  24469  3o1cs  28693  3o2cs  28694  poimirlem31  32610  tsan2  33119  frege114d  37069  ntrneiel2  37404  nnfoctbdjlem  39348