Theorem bj-bisimpl

Description: Implication from equivalence with a conjunct. Its associated inference is simplbi . (Contributed by BJ, 20-Mar-2026)

	Ref	Expression
Assertion	bj-bisimpl	$$⊢\left(\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)\to \left(\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biimp	$$⊢\left(\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)\to \left(\phi \to \left(\psi \wedge \chi \right)\right)$$
2		simpl	$$⊢\left(\psi \wedge \chi \right)\to \psi$$
3	1 2	syl6	$$⊢\left(\phi \leftrightarrow \left(\psi \wedge \chi \right)\right)\to \left(\phi \to \psi \right)$$