Theorem dfsb

Description: Simplify definition df-sb by removing its provable hypothesis. (Contributed by Wolf Lammen, 5-Feb-2026)

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

Proof

Step	Hyp	Ref	Expression
1		sbjust	$$⊢\forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\leftrightarrow \forall z\left(z=t\to \forall x\left(x=z\to \phi \right)\right)$$
2	1	df-sb	$$⊢\left[t/x\right]\phi \leftrightarrow \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$