Theorem nsb

Description: Any substitution in an always false formula is false. (Contributed by Steven Nguyen, 3-May-2023)

	Ref	Expression
Assertion	nsb	$$⊢\forall x\neg \phi \to \neg \left[t/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
2	1	biimpi	$$⊢\forall x\neg \phi \to \neg \exists x\phi$$
3		spsbe	$$⊢\left[t/x\right]\phi \to \exists x\phi$$
4	2 3	nsyl	$$⊢\forall x\neg \phi \to \neg \left[t/x\right]\phi$$