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Metamath Proof Explorer

Theorem imbi1i

Description: Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 17-Sep-2013)

		Ref	Expression
	Hypothesis	imbi1i.1	⊢ ( 𝜑 ↔ 𝜓 )
	Assertion	imbi1i	⊢ ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		imbi1i.1	⊢ ( 𝜑 ↔ 𝜓 )
2		imbi1	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝜑 → 𝜒 ) ↔ ( 𝜓 → 𝜒 ) )