MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  im2anan9r Structured version   Visualization version   GIF version

Theorem im2anan9r 877
Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
Hypotheses
Ref Expression
im2an9.1 (𝜑 → (𝜓𝜒))
im2an9.2 (𝜃 → (𝜏𝜂))
Assertion
Ref Expression
im2anan9r ((𝜃𝜑) → ((𝜓𝜏) → (𝜒𝜂)))

Proof of Theorem im2anan9r
StepHypRef Expression
1 im2an9.1 . . 3 (𝜑 → (𝜓𝜒))
2 im2an9.2 . . 3 (𝜃 → (𝜏𝜂))
31, 2im2anan9 876 . 2 ((𝜑𝜃) → ((𝜓𝜏) → (𝜒𝜂)))
43ancoms 468 1 ((𝜃𝜑) → ((𝜓𝜏) → (𝜒𝜂)))
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 196  df-an 385
This theorem is referenced by:  pssnn  8041  lbreu  10825  catideu  16108  wlkdvspthlem  25931  exidu1  32619  rngoideu  32666
  Copyright terms: Public domain W3C validator