This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem imbi2i

Description: Introduce an antecedent to both sides of a logical equivalence. This and the next three rules are useful for building up wff's around a definition, in order to make use of the definition. (Contributed by NM, 3-Jan-1993) (Proof shortened by Wolf Lammen, 6-Feb-2013)

		Ref	Expression
	Hypothesis	imbi2i.1	⊢ ( 𝜑 ↔ 𝜓 )
	Assertion	imbi2i	⊢ ( ( 𝜒 → 𝜑 ) ↔ ( 𝜒 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		imbi2i.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	a1i	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
3	2	pm5.74i	⊢ ( ( 𝜒 → 𝜑 ) ↔ ( 𝜒 → 𝜓 ) )