Theorem sbid

Description: An identity theorem for substitution. Remark 9.1 in Megill p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993) (Proof shortened by Wolf Lammen, 30-Sep-2018)

	Ref	Expression
Assertion	sbid	$$⊢\left[x/x\right]\phi \leftrightarrow \phi$$

Proof

Step	Hyp	Ref	Expression
1		equid	$$⊢x=x$$
2		sbequ12r	$$⊢x=x\to \left(\left[x/x\right]\phi \leftrightarrow \phi \right)$$
3	1 2	ax-mp	$$⊢\left[x/x\right]\phi \leftrightarrow \phi$$