Theorem dfsbimp

Description: A simple consequence of df-sb . (Contributed by Wolf Lammen, 4-Jun-2026)

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

Proof

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