Theorem sbi2

Description: Introduction of implication into substitution. (Contributed by NM, 14-May-1993)

	Ref	Expression
Assertion	sbi2	$$⊢\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\to \left[y/x\right]\left(\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbn	$$⊢\left[y/x\right]\neg \phi \leftrightarrow \neg \left[y/x\right]\phi$$
2		pm2.21	$$⊢\neg \phi \to \left(\phi \to \psi \right)$$
3	2	sbimi	$$⊢\left[y/x\right]\neg \phi \to \left[y/x\right]\left(\phi \to \psi \right)$$
4	1 3	sylbir	$$⊢\neg \left[y/x\right]\phi \to \left[y/x\right]\left(\phi \to \psi \right)$$
5		ax-1	$$⊢\psi \to \left(\phi \to \psi \right)$$
6	5	sbimi	$$⊢\left[y/x\right]\psi \to \left[y/x\right]\left(\phi \to \psi \right)$$
7	4 6	ja	$$⊢\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\to \left[y/x\right]\left(\phi \to \psi \right)$$