Theorem ichv

Description: Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023)

	Ref	Expression
Assertion	ichv	⊢ [ 𝑥 ⇄ 𝑦 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		sbv	⊢ ( [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
2	1	sbbii	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
3		sbv	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )
4	2 3	bitri	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
5	4	sbbii	⊢ ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑎 ] 𝜑 )
6		sbv	⊢ ( [ 𝑥 / 𝑎 ] 𝜑 ↔ 𝜑 )
7	5 6	bitri	⊢ ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
8	7	gen2	⊢ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
9		df-ich	⊢ ( [ 𝑥 ⇄ 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 ) )
10	8 9	mpbir	⊢ [ 𝑥 ⇄ 𝑦 ] 𝜑