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Theorem a1bi 350
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 348 . 2 (𝜑 → (𝜓 ↔ (𝜑𝜓)))
31, 2ax-mp 5 1 (𝜓 ↔ (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194
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 195
This theorem is referenced by:  mt2bi  351  pm4.83  965  truimfal  1505  equsalvw  1917  equsalhw  2108  equsal  2278  sbequ8ALT  2394  ralv  3191  relop  5182  acsfn0  16090  cmpsub  20955  ballotlemodife  29692  bj-trut  31546  bj-ssb1  31628  bj-equsalv  31737  bj-ralvw  31855  wl-equsald  32300  lub0N  33290  glb0N  33294
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