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Theorem nanbi1 1492
 Description: Introduce a right anti-conjunct to both sides of a logical equivalence. (Contributed by Anthony Hart, 1-Sep-2011.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
Assertion
Ref Expression
nanbi1 ((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Proof of Theorem nanbi1
StepHypRef Expression
1 imbi1 351 . 2 ((𝜑𝜓) → ((𝜑 → ¬ 𝜒) ↔ (𝜓 → ¬ 𝜒)))
2 nanimn 1485 . 2 ((𝜑𝜒) ↔ (𝜑 → ¬ 𝜒))
3 nanimn 1485 . 2 ((𝜓𝜒) ↔ (𝜓 → ¬ 𝜒))
41, 2, 33bitr4g 317 1 ((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ⊼ wnan 1482 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  df-an 400  df-nan 1483 This theorem is referenced by:  nanbi2  1493  nanbi12  1494  nanbi1i  1495  nanbi1d  1498
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