Theorem bian1d

Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017) (Proof shortened by Hongxiu Chen, 29-Jun-2025)

	Ref	Expression
Hypothesis	bian1d.1	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
Assertion	bian1d	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bian1d.1	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜒 ∧ 𝜃 ) ) )
2		ibar	⊢ ( 𝜒 → ( 𝜃 ↔ ( 𝜒 ∧ 𝜃 ) ) )
3	2	bicomd	⊢ ( 𝜒 → ( ( 𝜒 ∧ 𝜃 ) ↔ 𝜃 ) )
4	1 3	sylan9bb	⊢ ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ↔ 𝜃 ) )
5	4	ex	⊢ ( 𝜑 → ( 𝜒 → ( 𝜓 ↔ 𝜃 ) ) )
6	5	pm5.32d	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )