Theorem sbn1ALT

Description: Alternate proof of sbn1 , not using the false constant. (Contributed by BJ, 18-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sbn1ALT	$$⊢\left[t/x\right]\neg \phi \to \neg \left[t/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		nsb	$$⊢\forall x\neg \left(\phi \wedge \neg \phi \right)\to \neg \left[t/x\right]\left(\phi \wedge \neg \phi \right)$$
2		pm3.24	$$⊢\neg \left(\phi \wedge \neg \phi \right)$$
3	1 2	mpg	$$⊢\neg \left[t/x\right]\left(\phi \wedge \neg \phi \right)$$
4		sban	$$⊢\left[t/x\right]\left(\phi \wedge \neg \phi \right)\leftrightarrow \left(\left[t/x\right]\phi \wedge \left[t/x\right]\neg \phi \right)$$
5	3 4	mtbi	$$⊢\neg \left(\left[t/x\right]\phi \wedge \left[t/x\right]\neg \phi \right)$$
6		pm3.21	$$⊢\left[t/x\right]\neg \phi \to \left(\left[t/x\right]\phi \to \left(\left[t/x\right]\phi \wedge \left[t/x\right]\neg \phi \right)\right)$$
7	5 6	mtoi	$$⊢\left[t/x\right]\neg \phi \to \neg \left[t/x\right]\phi$$