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	$$⊢\phi \leftrightarrow \psi$$
Assertion	imbi1i	$$⊢\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		imbi1i.1	$$⊢\phi \leftrightarrow \psi$$
2		imbi1	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \chi \right)\right)$$
3	1 2	ax-mp	$$⊢\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \chi \right)$$