Theorem sb3

Description: One direction of a simplified definition of substitution when variables are distinct. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

	Ref	Expression
Assertion	sb3	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\to \left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb3b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)\right)$$
2	1	biimprd	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\to \left[y/x\right]\phi \right)$$