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Metamath Proof Explorer

Theorem anim12

Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d . Theorem *3.47 of WhiteheadRussell p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it praeclarum theorema (splendid theorem). (Contributed by NM, 12-Aug-1993) (Proof shortened by Wolf Lammen, 7-Apr-2013)

		Ref	Expression
	Assertion	anim12	$⊢ ((φ \to ψ) \land (χ \to θ)) \to ((φ \land χ) \to (ψ \land θ))$

Proof

Step	Hyp	Ref	Expression
1		id	$⊢ (φ \to ψ) \to (φ \to ψ)$
2		id	$⊢ (χ \to θ) \to (χ \to θ)$
3	1 2	im2anan9	$⊢ ((φ \to ψ) \land (χ \to θ)) \to ((φ \land χ) \to (ψ \land θ))$