Theorem sb1OLD

Description: Obsolete version of sb1 as of 21-Feb-2024. (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 29-Jul-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sb1OLD	$$⊢\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbequ2	$$⊢x=y\to \left(\left[y/x\right]\phi \to \phi \right)$$
2		19.8a	$$⊢\left(x=y\wedge \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
3	2	ex	$$⊢x=y\to \left(\phi \to \exists x\left(x=y\wedge \phi \right)\right)$$
4	1 3	syld	$$⊢x=y\to \left(\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)\right)$$
5	4	sps	$$⊢\forall xx=y\to \left(\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)\right)$$
6		sb4b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$
7		equs4	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$
8	6 7	syl6bi	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)\right)$$
9	5 8	pm2.61i	$$⊢\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)$$