Theorem ioin9i8 40101

Description: Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.)

Hypotheses

Ref	Expression
ioin9i8.1	⊢ (𝜑 → (𝜓 ∨ 𝜒))
ioin9i8.2	⊢ (𝜒 → ¬ 𝜃)
ioin9i8.3	⊢ (𝜓 → 𝜃)

Assertion

Ref	Expression
ioin9i8	⊢ (𝜑 → (𝜓 ↔ 𝜃))

Proof of Theorem ioin9i8

Step	Hyp	Ref	Expression
1		ioin9i8.3	. 2 ⊢ (𝜓 → 𝜃)
2		ioin9i8.1	. . . . 5 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
3	2	ord 860	. . . 4 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
4		ioin9i8.2	. . . 4 ⊢ (𝜒 → ¬ 𝜃)
5	3, 4	syl6 35	. . 3 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜃))
6	5	con4d 115	. 2 ⊢ (𝜑 → (𝜃 → 𝜓))
7	1, 6	impbid2 225	1 ⊢ (𝜑 → (𝜓 ↔ 𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843

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 206  df-or 844

This theorem is referenced by: (None)