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Metamath Proof Explorer

Theorem mpanr2

Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994) (Proof shortened by Andrew Salmon, 7-May-2011) (Proof shortened by Wolf Lammen, 7-Apr-2013)

	Ref	Expression
Hypotheses	mpanr2.1	⊢ 𝜒
	mpanr2.2	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) → 𝜃 )
Assertion	mpanr2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		mpanr2.1	⊢ 𝜒
2		mpanr2.2	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) → 𝜃 )
3	1	jctr	⊢ ( 𝜓 → ( 𝜓 ∧ 𝜒 ) )
4	3 2	sylan2	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )