Metamath Proof Explorer

Theorem bi2anan9

Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995)

	Ref	Expression
Hypotheses	bi2an9.1	$⊢ φ \to (ψ \leftrightarrow χ)$
	bi2an9.2	$⊢ θ \to (τ \leftrightarrow η)$
Assertion	bi2anan9	$⊢ (φ \land θ) \to ((ψ \land τ) \leftrightarrow (χ \land η))$

Step	Hyp	Ref	Expression
1		bi2an9.1	$⊢ φ \to (ψ \leftrightarrow χ)$
2		bi2an9.2	$⊢ θ \to (τ \leftrightarrow η)$
3		pm4.38	$⊢ ((ψ \leftrightarrow χ) \land (τ \leftrightarrow η)) \to ((ψ \land τ) \leftrightarrow (χ \land η))$
4	1 2 3	syl2an	$⊢ (φ \land θ) \to ((ψ \land τ) \leftrightarrow (χ \land η))$