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Theorem pm5.21nd 864
 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 114 . 2 (𝜑 → (𝜓𝜃))
3 pm5.21nd.2 . . 3 ((𝜑𝜒) → 𝜃)
43ex 114 . 2 (𝜑 → (𝜒𝜃))
5 pm5.21nd.3 . . 3 (𝜃 → (𝜓𝜒))
65a1i 9 . 2 (𝜑 → (𝜃 → (𝜓𝜒)))
72, 4, 6pm5.21ndd 657 1 (𝜑 → (𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  ideqg  4600  fvelimab  5373  releldm2  5969  relelec  6346  fzrev3  9562  elfzp12  9574  eltg  11813  eltg2  11814
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