Theorem sbralie

Description: Implicit to explicit substitution that swaps variables in a restrictedly universally quantified expression. (Contributed by NM, 5-Sep-2004) Avoid ax-ext , df-cleq , df-clel . (Revised by Wolf Lammen, 10-Mar-2025) Avoid ax-10 , ax-12 . (Revised by SN, 13-Nov-2025)

	Ref	Expression
Hypothesis	sbralie.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	sbralie	⊢ ( ∀ 𝑥 ∈ 𝑦 𝜑 ↔ [ 𝑦 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbralie.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝑦 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → 𝜑 ) )
3		elequ2	⊢ ( 𝑧 = 𝑦 → ( 𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦 ) )
4	3	imbi1d	⊢ ( 𝑧 = 𝑦 → ( ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ ( 𝑥 ∈ 𝑦 → 𝜑 ) ) )
5	4	sbievw	⊢ ( [ 𝑦 / 𝑧 ] ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ ( 𝑥 ∈ 𝑦 → 𝜑 ) )
6	5	albii	⊢ ( ∀ 𝑥 [ 𝑦 / 𝑧 ] ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → 𝜑 ) )
7		elequ1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) )
8	7 1	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ ( 𝑦 ∈ 𝑧 → 𝜓 ) ) )
9	8	cbvalvw	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝜓 ) )
10		df-ral	⊢ ( ∀ 𝑦 ∈ 𝑥 𝜓 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝜓 ) )
11	10	sbbii	⊢ ( [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 ↔ [ 𝑧 / 𝑥 ] ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝜓 ) )
12		sbal	⊢ ( [ 𝑧 / 𝑥 ] ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝜓 ) ↔ ∀ 𝑦 [ 𝑧 / 𝑥 ] ( 𝑦 ∈ 𝑥 → 𝜓 ) )
13		elequ2	⊢ ( 𝑥 = 𝑧 → ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧 ) )
14	13	imbi1d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑦 ∈ 𝑥 → 𝜓 ) ↔ ( 𝑦 ∈ 𝑧 → 𝜓 ) ) )
15	14	sbievw	⊢ ( [ 𝑧 / 𝑥 ] ( 𝑦 ∈ 𝑥 → 𝜓 ) ↔ ( 𝑦 ∈ 𝑧 → 𝜓 ) )
16	15	albii	⊢ ( ∀ 𝑦 [ 𝑧 / 𝑥 ] ( 𝑦 ∈ 𝑥 → 𝜓 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝜓 ) )
17	11 12 16	3bitrri	⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝑧 → 𝜓 ) ↔ [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 )
18	9 17	bitri	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 )
19	18	sbbii	⊢ ( [ 𝑦 / 𝑧 ] ∀ 𝑥 ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 )
20		sbal	⊢ ( [ 𝑦 / 𝑧 ] ∀ 𝑥 ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ ∀ 𝑥 [ 𝑦 / 𝑧 ] ( 𝑥 ∈ 𝑧 → 𝜑 ) )
21		sbco2vv	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 ↔ [ 𝑦 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 )
22	19 20 21	3bitr3i	⊢ ( ∀ 𝑥 [ 𝑦 / 𝑧 ] ( 𝑥 ∈ 𝑧 → 𝜑 ) ↔ [ 𝑦 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 )
23	2 6 22	3bitr2i	⊢ ( ∀ 𝑥 ∈ 𝑦 𝜑 ↔ [ 𝑦 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 𝜓 )