Theorem cbvabw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbvabw.1	⊢ Ⅎ 𝑦 𝜑
	cbvabw.2	⊢ Ⅎ 𝑥 𝜓
	cbvabw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvabw	⊢ { 𝑥 ∣ 𝜑 } = { 𝑦 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvabw.1	⊢ Ⅎ 𝑦 𝜑
2		cbvabw.2	⊢ Ⅎ 𝑥 𝜓
3		cbvabw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4		nfv	⊢ Ⅎ 𝑦 𝑥 = 𝑤
5	4 1	nfim	⊢ Ⅎ 𝑦 ( 𝑥 = 𝑤 → 𝜑 )
6		nfv	⊢ Ⅎ 𝑥 𝑦 = 𝑤
7	6 2	nfim	⊢ Ⅎ 𝑥 ( 𝑦 = 𝑤 → 𝜓 )
8		equequ1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑤 ↔ 𝑦 = 𝑤 ) )
9	8 3	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝑤 → 𝜑 ) ↔ ( 𝑦 = 𝑤 → 𝜓 ) ) )
10	5 7 9	cbvalv1	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜓 ) )
11	10	imbi2i	⊢ ( ( 𝑤 = 𝑧 → ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 ) ) ↔ ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜓 ) ) )
12	11	albii	⊢ ( ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 ) ) ↔ ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜓 ) ) )
13		df-sb	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑥 ( 𝑥 = 𝑤 → 𝜑 ) ) )
14		df-sb	⊢ ( [ 𝑧 / 𝑦 ] 𝜓 ↔ ∀ 𝑤 ( 𝑤 = 𝑧 → ∀ 𝑦 ( 𝑦 = 𝑤 → 𝜓 ) ) )
15	12 13 14	3bitr4i	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜓 )
16		df-clab	⊢ ( 𝑧 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑧 / 𝑥 ] 𝜑 )
17		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜓 } ↔ [ 𝑧 / 𝑦 ] 𝜓 )
18	15 16 17	3bitr4i	⊢ ( 𝑧 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑧 ∈ { 𝑦 ∣ 𝜓 } )
19	18	eqriv	⊢ { 𝑥 ∣ 𝜑 } = { 𝑦 ∣ 𝜓 }