Mathbox for Giovanni Mascellani < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  orel Structured version   Visualization version   GIF version

Theorem orel 35489
 Description: An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
Hypotheses
Ref Expression
orel.1 ((𝜓𝜂) → 𝜃)
orel.2 ((𝜒𝜌) → 𝜃)
orel.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
orel ((𝜑 ∧ (𝜂𝜌)) → 𝜃)

Proof of Theorem orel
StepHypRef Expression
1 simprl 770 . . 3 ((𝜑 ∧ (𝜂𝜌)) → 𝜂)
2 orel.1 . . . 4 ((𝜓𝜂) → 𝜃)
32ancoms 462 . . 3 ((𝜂𝜓) → 𝜃)
41, 3sylan 583 . 2 (((𝜑 ∧ (𝜂𝜌)) ∧ 𝜓) → 𝜃)
5 simprr 772 . . 3 ((𝜑 ∧ (𝜂𝜌)) → 𝜌)
6 orel.2 . . . 4 ((𝜒𝜌) → 𝜃)
76ancoms 462 . . 3 ((𝜌𝜒) → 𝜃)
85, 7sylan 583 . 2 (((𝜑 ∧ (𝜂𝜌)) ∧ 𝜒) → 𝜃)
9 orel.3 . . 3 (𝜑 → (𝜓𝜒))
109adantr 484 . 2 ((𝜑 ∧ (𝜂𝜌)) → (𝜓𝜒))
114, 8, 10mpjaodan 956 1 ((𝜑 ∧ (𝜂𝜌)) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844 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 210  df-an 400  df-or 845 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator