Theorem sbtvOLD

Description: Obsolete version of sbt as of 6-Jul-2023. A substitution into a theorem yields a theorem. See sbt when x , y need not be disjoint. (Contributed by BJ, 31-May-2019) Reduce axioms. (Revised by Steven Nguyen, 25-Apr-2023) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	sbtvOLD.1	$$⊢\phi$$
Assertion	sbtvOLD	$$⊢\left[x/y\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbtvOLD.1	$$⊢\phi$$
2	1	a1i	$$⊢y=x\to \phi$$
3		ax6ev	$$⊢\exists yy=x$$
4	3 1	exan	$$⊢\exists y\left(y=x\wedge \phi \right)$$
5		dfsb1	$$⊢\left[x/y\right]\phi \leftrightarrow \left(\left(y=x\to \phi \right)\wedge \exists y\left(y=x\wedge \phi \right)\right)$$
6	2 4 5	mpbir2an	$$⊢\left[x/y\right]\phi$$