Theorem sb4OLD

Description: Obsolete as of 30-Jul-2023. Use sb4b instead. One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993) Revise df-sb . (Revised by Wolf Lammen, 25-Jul-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sb4OLD	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \to \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sb4b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$
2	1	biimpd	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \to \forall x\left(x=y\to \phi \right)\right)$$