Theorem ichid

Description: A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023)

	Ref	Expression
Assertion	ichid	⊢ [ 𝑥 ⇄ 𝑥 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		sbid	⊢ ( [ 𝑥 / 𝑥 ] [ 𝑎 / 𝑥 ] 𝜑 ↔ [ 𝑎 / 𝑥 ] 𝜑 )
2	1	sbbii	⊢ ( [ 𝑥 / 𝑎 ] [ 𝑥 / 𝑥 ] [ 𝑎 / 𝑥 ] 𝜑 ↔ [ 𝑥 / 𝑎 ] [ 𝑎 / 𝑥 ] 𝜑 )
3		sbid2vw	⊢ ( [ 𝑥 / 𝑎 ] [ 𝑎 / 𝑥 ] 𝜑 ↔ 𝜑 )
4	2 3	bitri	⊢ ( [ 𝑥 / 𝑎 ] [ 𝑥 / 𝑥 ] [ 𝑎 / 𝑥 ] 𝜑 ↔ 𝜑 )
5	4	gen2	⊢ ∀ 𝑥 ∀ 𝑥 ( [ 𝑥 / 𝑎 ] [ 𝑥 / 𝑥 ] [ 𝑎 / 𝑥 ] 𝜑 ↔ 𝜑 )
6		df-ich	⊢ ( [ 𝑥 ⇄ 𝑥 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑥 ( [ 𝑥 / 𝑎 ] [ 𝑥 / 𝑥 ] [ 𝑎 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
7	5 6	mpbir	⊢ [ 𝑥 ⇄ 𝑥 ] 𝜑