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Theorem andiff 39790
Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.)
Hypotheses
Ref Expression
andiff.1 (𝜑 → (𝜒𝜃))
andiff.2 (𝜓 → (𝜃𝜒))
Assertion
Ref Expression
andiff ((𝜑𝜓) → (𝜒𝜃))

Proof of Theorem andiff
StepHypRef Expression
1 andiff.1 . . 3 (𝜑 → (𝜒𝜃))
2 andiff.2 . . 3 (𝜓 → (𝜃𝜒))
31, 2anim12i 616 . 2 ((𝜑𝜓) → ((𝜒𝜃) ∧ (𝜃𝜒)))
4 dfbi2 478 . 2 ((𝜒𝜃) ↔ ((𝜒𝜃) ∧ (𝜃𝜒)))
53, 4sylibr 237 1 ((𝜑𝜓) → (𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399
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
This theorem is referenced by: (None)
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