Theorem spsbimvOLD

Description: Obsolete version of spsbim as of 6-Jul-2023. Specialization of implication. Version of spsbim with a disjoint variable condition, not requiring ax-13 . (Contributed by Wolf Lammen, 19-Jan-2023) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall x\left(\phi \to \psi \right)$$
2		sp	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\phi \to \psi \right)$$
3	1 2	sbimd	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$