Theorem bian1d 580

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.)

Hypothesis

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

Assertion

Ref	Expression
bian1d	⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Proof of Theorem bian1d

Step	Hyp	Ref	Expression
1		bian1d.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
2	1	baibd 539	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃))
3	2	pm5.32da 579	1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  funcnvmpt  6944