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Theorem a1bi 351
 Description: Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
Hypothesis
Ref Expression
a1bi.1 𝜑
Assertion
Ref Expression
a1bi (𝜓 ↔ (𝜑𝜓))

Proof of Theorem a1bi
StepHypRef Expression
1 a1bi.1 . 2 𝜑
2 biimt 349 . 2 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
31, 2ax-mp 5 1 (𝜓 ↔ (𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195 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 196 This theorem is referenced by:  mt2bi  352  pm4.83  966  truimfal  1506  equsalvw  1918  equsalhw  2109  equsal  2279  sbequ8ALT  2395  ralv  3192  relop  5194  acsfn0  16144  cmpsub  21013  ballotlemodife  29886  bj-trut  31740  bj-ssb1  31822  bj-equsalv  31931  bj-ralvw  32059  wl-equsald  32504  lub0N  33494  glb0N  33498
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