Theorem bian1d

Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017)

	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	1	biimpd	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∧ 𝜃 ) ) )
3	2	adantld	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) → ( 𝜒 ∧ 𝜃 ) ) )
4		simpl	⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜒 )
5	4	a1i	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → 𝜒 ) )
6	1	biimprd	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → 𝜓 ) )
7	5 6	jcad	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜃 ) → ( 𝜒 ∧ 𝜓 ) ) )
8	3 7	impbid	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )