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Theorem pm5.21nd 868
 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 ((φ ψ) → θ)
pm5.21nd.2 ((φ χ) → 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 ((φ ψ) → θ)
21ex 423 . 2 (φ → (ψθ))
3 pm5.21nd.2 . . 3 ((φ χ) → θ)
43ex 423 . 2 (φ → (χθ))
5 pm5.21nd.3 . . 3 (θ → (ψχ))
65a1i 10 . 2 (φ → (θ → (ψχ)))
72, 4, 6pm5.21ndd 343 1 (φ → (ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358 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 177  df-an 360 This theorem is referenced by:  ideqg  4868  ideqg2  4869  fvelimab  5370
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