Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  a2and Structured version   Visualization version   GIF version

Theorem a2and 870
 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 451 . . . . . 6 (𝜑 → (𝜓 → (𝜌𝜒)))
32imdistand 728 . . . . 5 (𝜑 → ((𝜓𝜌) → (𝜓𝜒)))
43imp 444 . . . 4 ((𝜑 ∧ (𝜓𝜌)) → (𝜓𝜒))
5 a2and.1 . . . . 5 (𝜑 → ((𝜓𝜌) → (𝜏𝜃)))
65imp 444 . . . 4 ((𝜑 ∧ (𝜓𝜌)) → (𝜏𝜃))
74, 6embantd 59 . . 3 ((𝜑 ∧ (𝜓𝜌)) → (((𝜓𝜒) → 𝜏) → 𝜃))
87ex 449 . 2 (𝜑 → ((𝜓𝜌) → (((𝜓𝜒) → 𝜏) → 𝜃)))
98com23 86 1 (𝜑 → (((𝜓𝜒) → 𝜏) → ((𝜓𝜌) → 𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ 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 197  df-an 385 This theorem is referenced by:  telgsumfzs  18432
 Copyright terms: Public domain W3C validator