Theorem jao 975

Description: Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.)

Assertion

Ref	Expression
jao	⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((𝜑 ∨ 𝜒) → 𝜓)))

Proof of Theorem jao

Step	Hyp	Ref	Expression
1		pm3.44 974	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) → ((𝜑 ∨ 𝜒) → 𝜓))
2	1	ex 418	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((𝜑 ∨ 𝜒) → 𝜓)))

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-an 402  df-or 862

This theorem is used by:  3jao  1452  en3lplem2  9598  indpi  10973  axtco2  37232  bj-orim2  37395  jaodd  43228  jaoded  45508  suctrALT2VD  45777  suctrALT2  45778  en3lplem2VD  45785  hbimpgVD  45845  ax6e2ndeqVD  45850  suctrALTcf  45863  suctrALTcfVD  45864  ax6e2ndeqALT  45872