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Metamath Proof Explorer

Theorem simpl2im

Description: Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021) (Proof shortened by Wolf Lammen, 26-Mar-2022)

	Ref	Expression
Hypotheses	simpl2im.1	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )
	simpl2im.2	⊢ ( 𝜒 → 𝜃 )
Assertion	simpl2im	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		simpl2im.1	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )
2		simpl2im.2	⊢ ( 𝜒 → 𝜃 )
3	1	simprd	⊢ ( 𝜑 → 𝜒 )
4	3 2	syl	⊢ ( 𝜑 → 𝜃 )