Theorem sb3bOLD

Description: Obsolete version of sb3b as of 21-Feb-2024. (Contributed by BJ, 6-Oct-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sb3bOLD	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sb1	$$⊢\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)$$
2		sb3	$$⊢\neg \forall xx=y\to \left(\exists x\left(x=y\wedge \phi \right)\to \left[y/x\right]\phi \right)$$
3	1 2	impbid2	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)\right)$$