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Theorem pm5.21nd 801
Description: Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 379. (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 412 . 2 (𝜑 → (𝜓𝜃))
3 pm5.21nd.2 . . 3 ((𝜑𝜒) → 𝜃)
43ex 412 . 2 (𝜑 → (𝜒𝜃))
5 pm5.21nd.3 . . 3 (𝜃 → (𝜓𝜒))
65a1i 11 . 2 (𝜑 → (𝜃 → (𝜓𝜒)))
72, 4, 6pm5.21ndd 379 1 (𝜑 → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395
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 206  df-an 396
This theorem is referenced by:  ideqg  5854  fvelimab  6971  brrpssg  7730  ordsucelsuc  7825  releldm2  8047  relbrtpos  8243  relelec  8771  elfiun  9454  fpwwe2lem2  10656  fpwwelem  10669  fzrev3  13600  elfzp12  13613  eqgval  19132  eltg  22873  eltg2  22874  cncnp2  23198  isref  23426  islocfin  23434  opeldifid  32402  isfne  35823  copsex2b  36619  bj-ideqgALT  36637  bj-idreseq  36641  bj-ideqg1ALT  36644  opelopab3  37191  isdivrngo  37423  brssr  37973  islshpkrN  38592  dihatexv2  40812
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