Theorem wl-sblimt

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim . (Contributed by Wolf Lammen, 26-Jul-2019)

	Ref	Expression
Assertion	wl-sblimt	$$⊢Ⅎx\psi \to \left(\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sbim	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$
2		sbft	$$⊢Ⅎx\psi \to \left(\left[y/x\right]\psi \leftrightarrow \psi \right)$$
3	2	imbi2d	$$⊢Ⅎx\psi \to \left(\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \psi \right)\right)$$
4	1 3	bitrid	$$⊢Ⅎx\psi \to \left(\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \psi \right)\right)$$