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Theorem bi123imp0 41136
Description: Similar to 3imp 1108 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
Hypothesis
Ref Expression
bi23imp0.1 (𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))
Assertion
Ref Expression
bi123imp0 ((𝜑𝜓𝜒) → 𝜃)

Proof of Theorem bi123imp0
StepHypRef Expression
1 bi23imp0.1 . . 3 (𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))
2 biimp 218 . . . 4 ((𝜓 ↔ (𝜒𝜃)) → (𝜓 → (𝜒𝜃)))
3 biimp 218 . . . 4 ((𝜒𝜃) → (𝜒𝜃))
42, 3syl6 35 . . 3 ((𝜓 ↔ (𝜒𝜃)) → (𝜓 → (𝜒𝜃)))
51, 4sylbi 220 . 2 (𝜑 → (𝜓 → (𝜒𝜃)))
653imp 1108 1 ((𝜑𝜓𝜒) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084
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-3an 1086
This theorem is referenced by: (None)
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