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Theorem bi2bian9 550
Description: Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
Hypotheses
Ref Expression
bi2an9.1 (𝜑 → (𝜓𝜒))
bi2an9.2 (𝜃 → (𝜏𝜂))
Assertion
Ref Expression
bi2bian9 ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))

Proof of Theorem bi2bian9
StepHypRef Expression
1 bi2an9.1 . . 3 (𝜑 → (𝜓𝜒))
21adantr 265 . 2 ((𝜑𝜃) → (𝜓𝜒))
3 bi2an9.2 . . 3 (𝜃 → (𝜏𝜂))
43adantl 266 . 2 ((𝜑𝜃) → (𝜏𝜂))
52, 4bibi12d 228 1 ((𝜑𝜃) → ((𝜓𝜏) ↔ (𝜒𝜂)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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