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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 → 𝜓 ) ∧ ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ∧ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) )
2		id	⊢ ( ( 𝜒 → 𝜃 ) → ( 𝜒 → 𝜃 ) )
3	1 2	im2anan9	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜃 ) ) → ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ∧ 𝜃 ) ) )