Theorem spsbeOLD

Description: Obsolete version of spsbe as of 11-Jul-2023. (Contributed by NM, 29-Jun-1993) (Proof shortened by Wolf Lammen, 3-May-2018) Revise df-sb . (Revised by BJ, 22-Dec-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	spsbeOLD	$$⊢\left[t/x\right]\phi \to \exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-sb	$$⊢\left[t/x\right]\phi \leftrightarrow \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
2		ax6ev	$$⊢\exists yy=t$$
3		exim	$$⊢\forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\to \left(\exists yy=t\to \exists y\forall x\left(x=y\to \phi \right)\right)$$
4	2 3	mpi	$$⊢\forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\to \exists y\forall x\left(x=y\to \phi \right)$$
5	1 4	sylbi	$$⊢\left[t/x\right]\phi \to \exists y\forall x\left(x=y\to \phi \right)$$
6		exim	$$⊢\forall x\left(x=y\to \phi \right)\to \left(\exists xx=y\to \exists x\phi \right)$$
7	6	eximi	$$⊢\exists y\forall x\left(x=y\to \phi \right)\to \exists y\left(\exists xx=y\to \exists x\phi \right)$$
8		ax6ev	$$⊢\exists xx=y$$
9		pm2.27	$$⊢\exists xx=y\to \left(\left(\exists xx=y\to \exists x\phi \right)\to \exists x\phi \right)$$
10	8 9	ax-mp	$$⊢\left(\exists xx=y\to \exists x\phi \right)\to \exists x\phi$$
11	10	exlimiv	$$⊢\exists y\left(\exists xx=y\to \exists x\phi \right)\to \exists x\phi$$
12	5 7 11	3syl	$$⊢\left[t/x\right]\phi \to \exists x\phi$$