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

Theorem jaoa 531
 Description: Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
Hypotheses
Ref Expression
jaao.1 (𝜑 → (𝜓𝜒))
jaao.2 (𝜃 → (𝜏𝜒))
Assertion
Ref Expression
jaoa ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Proof of Theorem jaoa
StepHypRef Expression
1 jaao.1 . . 3 (𝜑 → (𝜓𝜒))
21adantrd 483 . 2 (𝜑 → ((𝜓𝜏) → 𝜒))
3 jaao.2 . . 3 (𝜃 → (𝜏𝜒))
43adantld 482 . 2 (𝜃 → ((𝜓𝜏) → 𝜒))
52, 4jaoi 393 1 ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))
 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:  pm4.79  605  19.40b  1804  abslt  13902  absle  13903  uncon  21042  dfon2lem4  30935  clsk1indlem3  37361
 Copyright terms: Public domain W3C validator