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Theorem orel 32868
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 790 . . 3 ((𝜑 ∧ (𝜂𝜌)) → 𝜂)
2 orel.1 . . . 4 ((𝜓𝜂) → 𝜃)
32ancoms 468 . . 3 ((𝜂𝜓) → 𝜃)
41, 3sylan 487 . 2 (((𝜑 ∧ (𝜂𝜌)) ∧ 𝜓) → 𝜃)
5 simprr 792 . . 3 ((𝜑 ∧ (𝜂𝜌)) → 𝜌)
6 orel.2 . . . 4 ((𝜒𝜌) → 𝜃)
76ancoms 468 . . 3 ((𝜌𝜒) → 𝜃)
85, 7sylan 487 . 2 (((𝜑 ∧ (𝜂𝜌)) ∧ 𝜒) → 𝜃)
9 orel.3 . . 3 (𝜑 → (𝜓𝜒))
109adantr 480 . 2 ((𝜑 ∧ (𝜂𝜌)) → (𝜓𝜒))
114, 8, 10mpjaodan 823 1 ((𝜑 ∧ (𝜂𝜌)) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383
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  df-an 385
This theorem is referenced by: (None)
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