Theorem sbim

Description: Implication inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993)

	Ref	Expression
Assertion	sbim	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbi1	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$
2		sbi2	$$⊢\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\to \left[y/x\right]\left(\phi \to \psi \right)$$
3	1 2	impbii	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$