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Theorem impbi 211
 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 210 . . 3 ¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
2 simprim 169 . . 3 (¬ (((𝜑𝜓) → ¬ ((𝜑𝜓) → ¬ (𝜓𝜑))) → ¬ (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))) → (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓)))
31, 2ax-mp 5 . 2 (¬ ((𝜑𝜓) → ¬ (𝜓𝜑)) → (𝜑𝜓))
43expi 168 1 ((𝜑𝜓) → ((𝜓𝜑) → (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209 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 210 This theorem is referenced by:  impbii  212  impbidd  213  impbid21d  214  dfbi1  216  impimprbi  827  bj-bisym  34056  bj-moeub  34307  eqsbc3rVD  41589  orbi1rVD  41597  3impexpVD  41605  3impexpbicomVD  41606  imbi12VD  41622  sbcim2gVD  41624  sb5ALTVD  41662
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