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Metamath Proof Explorer

Theorem con3

Description: Contraposition. Theorem *2.16 of WhiteheadRussell p. 103. This was the fourth axiom of Frege, specifically Proposition 28 of Frege1879 p. 43. Its associated inference is con3i . (Contributed by NM, 29-Dec-1992) (Proof shortened by Wolf Lammen, 13-Feb-2013)

		Ref	Expression
	Assertion	con3	⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) )
2	1	con3d	⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ 𝜑 ) )