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Theorem pm5.21nd 940
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 450 . 2 (𝜑 → (𝜓𝜃))
3 pm5.21nd.2 . . 3 ((𝜑𝜒) → 𝜃)
43ex 450 . 2 (𝜑 → (𝜒𝜃))
5 pm5.21nd.3 . . 3 (𝜃 → (𝜓𝜒))
65a1i 11 . 2 (𝜑 → (𝜃 → (𝜓𝜒)))
72, 4, 6pm5.21ndd 369 1 (𝜑 → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384
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 386
This theorem is referenced by:  ideqg  5233  fvelimab  6210  brrpssg  6892  ordsucelsuc  6969  releldm2  7163  relbrtpos  7308  relelec  7732  elfiun  8280  fpwwe2lem2  9398  fpwwelem  9411  fzrev3  12348  elfzp12  12360  eqgval  17564  eltg  20672  eltg2  20673  cncnp2  20995  isref  21222  islocfin  21230  opeldifid  29254  isfne  31973  opelopab3  33140  isdivrngo  33378  islshpkrN  33884  dihatexv2  36105
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