Theorem sb1

Description: One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition ( sb5 ) or a nonfreeness hypothesis ( sb5f ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sb1v when possible. (Contributed by NM, 13-May-1993) Revise df-sb . (Revised by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

	Ref	Expression
Assertion	sb1	$$⊢\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		spsbe	$$⊢\left[y/x\right]\phi \to \exists x\phi$$
2		pm3.2	$$⊢x=y\to \left(\phi \to \left(x=y\wedge \phi \right)\right)$$
3	2	aleximi	$$⊢\forall xx=y\to \left(\exists x\phi \to \exists x\left(x=y\wedge \phi \right)\right)$$
4	1 3	syl5	$$⊢\forall xx=y\to \left(\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)\right)$$
5		sb3b	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)\right)$$
6	5	biimpd	$$⊢\neg \forall xx=y\to \left(\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)\right)$$
7	4 6	pm2.61i	$$⊢\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)$$