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Metamath Proof Explorer

Theorem impexp

Description: Import-export theorem. Part of Theorem *4.87 of WhiteheadRussell p. 122. (Contributed by NM, 10-Jan-1993) (Proof shortened by Wolf Lammen, 24-Mar-2013)

		Ref	Expression
	Assertion	impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm3.3	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) → ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
2		pm3.31	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) )
3	1 2	impbii	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( 𝜑 → ( 𝜓 → 𝜒 ) ) )