Theorem ichf

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

	Ref	Expression
Hypotheses	ichf.1	⊢ Ⅎ 𝑥 𝜑
	ichf.2	⊢ Ⅎ 𝑦 𝜑
Assertion	ichf	⊢ [ 𝑥 ⇄ 𝑦 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		ichf.1	⊢ Ⅎ 𝑥 𝜑
2		ichf.2	⊢ Ⅎ 𝑦 𝜑
3	2	sbf	⊢ ( [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
4	3	sbbii	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
5	1	sbf	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )
6	4 5	bitri	⊢ ( [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
7	6	sbbii	⊢ ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑎 ] 𝜑 )
8		sbv	⊢ ( [ 𝑥 / 𝑎 ] 𝜑 ↔ 𝜑 )
9	7 8	bitri	⊢ ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
10	9	gen2	⊢ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 )
11		df-ich	⊢ ( [ 𝑥 ⇄ 𝑦 ] 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜑 ↔ 𝜑 ) )
12	10 11	mpbir	⊢ [ 𝑥 ⇄ 𝑦 ] 𝜑