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Metamath Proof Explorer

Theorem biimprd

Description: Deduce a converse implication from a logical equivalence. Deduction associated with biimpr and biimpri . (Contributed by NM, 11-Jan-1993) (Proof shortened by Wolf Lammen, 22-Sep-2013)

		Ref	Expression
	Hypothesis	biimprd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	Assertion	biimprd	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		biimprd.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		id	⊢ ( 𝜒 → 𝜒 )
3	2 1	imbitrrid	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )