Theorem sblim

Description: Substitution in an implication with a variable not free in the consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013) (Revised by Mario Carneiro, 4-Oct-2016)

	Ref	Expression
Hypothesis	sblim.1	$$⊢Ⅎx\psi$$
Assertion	sblim	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sblim.1	$$⊢Ⅎx\psi$$
2		sbim	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$
3	1	sbf	$$⊢\left[y/x\right]\psi \leftrightarrow \psi$$
4	3	imbi2i	$$⊢\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \psi \right)$$
5	2 4	bitri	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \psi \right)$$