Theorem sb2

Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition ( sb6 ) or a non-freeness hypothesis ( sb6f ). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 26-Jul-2023) (New usage is discouraged.)

	Ref	Expression
Assertion	sb2	$$⊢\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		pm2.27	$$⊢x=y\to \left(\left(x=y\to \phi \right)\to \phi \right)$$
2	1	al2imi	$$⊢\forall xx=y\to \left(\forall x\left(x=y\to \phi \right)\to \forall x\phi \right)$$
3		stdpc4	$$⊢\forall x\phi \to \left[y/x\right]\phi$$
4	2 3	syl6	$$⊢\forall xx=y\to \left(\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi \right)$$
5		sb4b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)\right)$$
6	5	biimprd	$$⊢\neg \forall xx=y\to \left(\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi \right)$$
7	4 6	pm2.61i	$$⊢\forall x\left(x=y\to \phi \right)\to \left[y/x\right]\phi$$