Theorem spsbim

Description: Distribute substitution over implication. Closed form of sbimi . Specialization of implication. (Contributed by NM, 5-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011) Revise df-sb . (Revised by BJ, 22-Dec-2020) (Proof shortened by Steven Nguyen, 24-Jul-2023)

	Ref	Expression
Assertion	spsbim	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\left[t/x\right]\phi \to \left[t/x\right]\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		stdpc4	$$⊢\forall x\left(\phi \to \psi \right)\to \left[t/x\right]\left(\phi \to \psi \right)$$
2		sbi1	$$⊢\left[t/x\right]\left(\phi \to \psi \right)\to \left(\left[t/x\right]\phi \to \left[t/x\right]\psi \right)$$
3	1 2	syl	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\left[t/x\right]\phi \to \left[t/x\right]\psi \right)$$