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Theorem impbi 198
Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
Assertion
Ref Expression
impbi ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))

Proof of Theorem impbi
StepHypRef Expression
1 df-bi 197 . . 3 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
2 simprim 163 . . 3 (¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))) → (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
31, 2ax-mp 5 . 2 (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))
43expi 162 1 ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196
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
This theorem is referenced by:  impbii  199  impbidd  200  dfbi1  203  bj-bisym  32912  eqsbc3rVD  39597  orbi1rVD  39605  3impexpVD  39613  3impexpbicomVD  39614  imbi12VD  39631  sbcim2gVD  39633  sb5ALTVD  39671
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