Theorem orcs 889

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 881	. 2 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2		orcs.1	. 2 ⊢ ((𝜑 ∨ 𝜓) → 𝜒)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 862

This theorem is used by:  olcs  890  norasslem2  1565  ifor  4540  tppreqb  4768  frxp  8165  mndifsplit  22981  maducoeval2  22985  leibpilem2  27299  leibpi  27300  3o1cs  33131  3o2cs  33132  elrgspnlem2  33874  poimirlem31  38761  tsan2  39266  frege114d  44950  ntrneiel2  45278  nnfoctbdjlem  47670  homf0  50328