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Theorem pm5.21nd 961
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 196  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:  ideqg  5306  fvelimab  6292  brrpssg  6981  ordsucelsuc  7064  releldm2  7262  relbrtpos  7408  relelec  7830  elfiun  8377  fpwwe2lem2  9492  fpwwelem  9505  fzrev3  12444  elfzp12  12457  eqgval  17690  eltg  20809  eltg2  20810  cncnp2  21133  isref  21360  islocfin  21368  opeldifid  29538  isfne  32459  opelopab3  33641  isdivrngo  33879  brssr  34391  islshpkrN  34725  dihatexv2  36945
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