Theorem sbrim

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. See sbrimv for a version with disjoint variables not requiring ax-10 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016)

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

Proof

Step	Hyp	Ref	Expression
1		sbrim.1	$$⊢Ⅎx\phi$$
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	$$⊢\left[y/x\right]\phi \leftrightarrow \phi$$
4	3	imbi1i	$$⊢\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\leftrightarrow \left(\phi \to \left[y/x\right]\psi \right)$$
5	2 4	bitri	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \left[y/x\right]\psi \right)$$