This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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) (Proof shortened by Peter Mazsa, 24-Feb-2026)

		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	baibd	⊢ ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ↔ 𝜃 ) )
3	2	pm5.32da	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) ↔ ( 𝜒 ∧ 𝜃 ) ) )