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Metamath Proof Explorer

Theorem eelT1

Description: Syllogism inference combined withmodus ponens. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Alan Sare, 23-Dec-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	eelT1.1	⊢ ( ⊤ → 𝜑 )
	eelT1.2	⊢ ( 𝜓 → 𝜒 )
	eelT1.3	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
Assertion	eelT1	⊢ ( 𝜓 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		eelT1.1	⊢ ( ⊤ → 𝜑 )
2		eelT1.2	⊢ ( 𝜓 → 𝜒 )
3		eelT1.3	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
4	1	mptru	⊢ 𝜑
5	4 2 3	sylancr	⊢ ( 𝜓 → 𝜃 )