Theorem sbimvOLD

Description: Obsolete version of sbim as of 24-Jul-2023. Substitution distributes over implication. Version of sbim with a disjoint variable condition, not requiring ax-13 . (Contributed by Wolf Lammen, 18-Jan-2023) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sbi1vOLD	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$
2		sbi2	$$⊢\left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\to \left[y/x\right]\left(\phi \to \psi \right)$$
3	1 2	impbii	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$