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Theorem a2and 842
 Description: Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
Hypotheses
Ref Expression
a2and.1 (𝜑 → ((𝜓𝜌) → (𝜏𝜃)))
a2and.2 (𝜑 → ((𝜓𝜌) → 𝜒))
Assertion
Ref Expression
a2and (𝜑 → (((𝜓𝜒) → 𝜏) → ((𝜓𝜌) → 𝜃)))

Proof of Theorem a2and
StepHypRef Expression
1 a2and.1 . . 3 (𝜑 → ((𝜓𝜌) → (𝜏𝜃)))
2 a2and.2 . . . . 5 (𝜑 → ((𝜓𝜌) → 𝜒))
32expd 419 . . . 4 (𝜑 → (𝜓 → (𝜌𝜒)))
43imdistand 574 . . 3 (𝜑 → ((𝜓𝜌) → (𝜓𝜒)))
5 imim1 83 . . . 4 (((𝜓𝜒) → 𝜏) → ((𝜏𝜃) → ((𝜓𝜒) → 𝜃)))
65com3l 89 . . 3 ((𝜏𝜃) → ((𝜓𝜒) → (((𝜓𝜒) → 𝜏) → 𝜃)))
71, 4, 6syl6c 70 . 2 (𝜑 → ((𝜓𝜌) → (((𝜓𝜒) → 𝜏) → 𝜃)))
87com23 86 1 (𝜑 → (((𝜓𝜒) → 𝜏) → ((𝜓𝜌) → 𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399 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 This theorem is referenced by:  telgsumfzs  19105
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