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Theorem ioin9i8 39708
 Description: Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.)
Hypotheses
Ref Expression
ioin9i8.1 (𝜑 → (𝜓𝜒))
ioin9i8.2 (𝜒 → ¬ 𝜃)
ioin9i8.3 (𝜓𝜃)
Assertion
Ref Expression
ioin9i8 (𝜑 → (𝜓𝜃))

Proof of Theorem ioin9i8
StepHypRef Expression
1 ioin9i8.3 . 2 (𝜓𝜃)
2 ioin9i8.1 . . . . 5 (𝜑 → (𝜓𝜒))
32ord 861 . . . 4 (𝜑 → (¬ 𝜓𝜒))
4 ioin9i8.2 . . . 4 (𝜒 → ¬ 𝜃)
53, 4syl6 35 . . 3 (𝜑 → (¬ 𝜓 → ¬ 𝜃))
65con4d 115 . 2 (𝜑 → (𝜃𝜓))
71, 6impbid2 229 1 (𝜑 → (𝜓𝜃))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 844 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-or 845 This theorem is referenced by: (None)
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