Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm5.21nd Structured version   Visualization version   GIF version

Theorem pm5.21nd 939
 Description: Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
Hypotheses
Ref Expression
pm5.21nd.1 ((𝜑𝜓) → 𝜃)
pm5.21nd.2 ((𝜑𝜒) → 𝜃)
pm5.21nd.3 (𝜃 → (𝜓𝜒))
Assertion
Ref Expression
pm5.21nd (𝜑 → (𝜓𝜒))

Proof of Theorem pm5.21nd
StepHypRef Expression
1 pm5.21nd.1 . . 3 ((𝜑𝜓) → 𝜃)
21ex 449 . 2 (𝜑 → (𝜓𝜃))
3 pm5.21nd.2 . . 3 ((𝜑𝜒) → 𝜃)
43ex 449 . 2 (𝜑 → (𝜒𝜃))
5 pm5.21nd.3 . . 3 (𝜃 → (𝜓𝜒))
65a1i 11 . 2 (𝜑 → (𝜃 → (𝜓𝜒)))
72, 4, 6pm5.21ndd 368 1 (𝜑 → (𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ 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:  ideqg  5195  fvelimab  6163  brrpssg  6837  ordsucelsuc  6914  releldm2  7109  relbrtpos  7250  relelec  7674  elfiun  8219  fpwwe2lem2  9333  fpwwelem  9346  fzrev3  12276  elfzp12  12288  eqgval  17466  eltg  20572  eltg2  20573  cncnp2  20895  isref  21122  islocfin  21130  opeldifid  28794  isfne  31504  opelopab3  32681  isdivrngo  32919  islshpkrN  33425  dihatexv2  35646
 Copyright terms: Public domain W3C validator