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Theorem bijadc 850
Description: Combine antecedents into a single biconditional. This inference is reminiscent of jadc 831. (Contributed by Jim Kingdon, 4-May-2018.)
Hypotheses
Ref Expression
bijadc.1 (𝜑 → (𝜓𝜒))
bijadc.2 𝜑 → (¬ 𝜓𝜒))
Assertion
Ref Expression
bijadc (DECID 𝜓 → ((𝜑𝜓) → 𝜒))

Proof of Theorem bijadc
StepHypRef Expression
1 bi2 129 . . 3 ((𝜑𝜓) → (𝜓𝜑))
2 bijadc.1 . . 3 (𝜑 → (𝜓𝜒))
31, 2syli 37 . 2 ((𝜑𝜓) → (𝜓𝜒))
4 bi1 117 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
54con3d 603 . . 3 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
6 bijadc.2 . . 3 𝜑 → (¬ 𝜓𝜒))
75, 6syli 37 . 2 ((𝜑𝜓) → (¬ 𝜓𝜒))
83, 7pm2.61ddc 829 1 (DECID 𝜓 → ((𝜑𝜓) → 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  DECID wdc 802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803
This theorem is referenced by: (None)
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