Theorem sbequ2

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993) Revise df-sb . (Revised by BJ, 22-Dec-2020) (Proof shortened by Wolf Lammen, 3-Feb-2024)

	Ref	Expression
Assertion	sbequ2	$$⊢x=t\to \left(\left[t/x\right]\phi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-sb	$$⊢\left[t/x\right]\phi \leftrightarrow \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
2	1	biimpi	$$⊢\left[t/x\right]\phi \to \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
3		equvinva	$$⊢x=t\to \exists y\left(x=y\wedge t=y\right)$$
4		equcomi	$$⊢t=y\to y=t$$
5		sp	$$⊢\forall x\left(x=y\to \phi \right)\to \left(x=y\to \phi \right)$$
6	4 5	imim12i	$$⊢\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\to \left(t=y\to \left(x=y\to \phi \right)\right)$$
7	6	impcomd	$$⊢\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\to \left(\left(x=y\wedge t=y\right)\to \phi \right)$$
8	7	aleximi	$$⊢\forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\to \left(\exists y\left(x=y\wedge t=y\right)\to \exists y\phi \right)$$
9	2 3 8	syl2im	$$⊢\left[t/x\right]\phi \to \left(x=t\to \exists y\phi \right)$$
10		ax5e	$$⊢\exists y\phi \to \phi$$
11	9 10	syl6com	$$⊢x=t\to \left(\left[t/x\right]\phi \to \phi \right)$$