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Theorem a2and 526
 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.2 . . . . . . 7 (𝜑 → ((𝜓𝜌) → 𝜒))
21expd 255 . . . . . 6 (𝜑 → (𝜓 → (𝜌𝜒)))
32imdistand 437 . . . . 5 (𝜑 → ((𝜓𝜌) → (𝜓𝜒)))
43imp 123 . . . 4 ((𝜑 ∧ (𝜓𝜌)) → (𝜓𝜒))
5 a2and.1 . . . . 5 (𝜑 → ((𝜓𝜌) → (𝜏𝜃)))
65imp 123 . . . 4 ((𝜑 ∧ (𝜓𝜌)) → (𝜏𝜃))
74, 6embantd 56 . . 3 ((𝜑 ∧ (𝜓𝜌)) → (((𝜓𝜒) → 𝜏) → 𝜃))
87ex 114 . 2 (𝜑 → ((𝜓𝜌) → (((𝜓𝜒) → 𝜏) → 𝜃)))
98com23 78 1 (𝜑 → (((𝜓𝜒) → 𝜏) → ((𝜓𝜌) → 𝜃)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
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