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	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		equid	⊢ 𝑥 = 𝑥
2		sbequ12r	⊢ ( 𝑥 = 𝑥 → ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	1 2	ax-mp	⊢ ( [ 𝑥 / 𝑥 ] 𝜑 ↔ 𝜑 )