Theorem cbvralfw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvralf with a disjoint variable condition, which does not require ax-10 , ax-13 . For a version not dependent on ax-11 and ax-12, see cbvralvw . (Contributed by NM, 7-Mar-2004) Avoid ax-10 , ax-13 . (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbvralfw.1	⊢ Ⅎ 𝑥 𝐴
	cbvralfw.2	⊢ Ⅎ 𝑦 𝐴
	cbvralfw.3	⊢ Ⅎ 𝑦 𝜑
	cbvralfw.4	⊢ Ⅎ 𝑥 𝜓
	cbvralfw.5	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvralfw	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvralfw.1	⊢ Ⅎ 𝑥 𝐴
2		cbvralfw.2	⊢ Ⅎ 𝑦 𝐴
3		cbvralfw.3	⊢ Ⅎ 𝑦 𝜑
4		cbvralfw.4	⊢ Ⅎ 𝑥 𝜓
5		cbvralfw.5	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
6	2	nfcri	⊢ Ⅎ 𝑦 𝑥 ∈ 𝐴
7	6 3	nfim	⊢ Ⅎ 𝑦 ( 𝑥 ∈ 𝐴 → 𝜑 )
8	1	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴
9	8 4	nfim	⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 → 𝜓 )
10		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
11	10 5	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → 𝜓 ) ) )
12	7 9 11	cbvalv1	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
13		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
14		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
15	12 13 14	3bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 )