Theorem cbvreuw

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 15-Oct-2016) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024) Avoid ax-10 . (Revised by Wolf Lammen, 10-Dec-2024)

	Ref	Expression
Hypotheses	cbvreuw.1	⊢ Ⅎ 𝑦 𝜑
	cbvreuw.2	⊢ Ⅎ 𝑥 𝜓
	cbvreuw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvreuw	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvreuw.1	⊢ Ⅎ 𝑦 𝜑
2		cbvreuw.2	⊢ Ⅎ 𝑥 𝜓
3		cbvreuw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1 2 3	cbvrexw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 𝜓 )
5	1 2 3	cbvrmow	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑦 ∈ 𝐴 𝜓 )
6	4 5	anbi12i	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝜓 ∧ ∃* 𝑦 ∈ 𝐴 𝜓 ) )
7		reu5	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 𝜑 ) )
8		reu5	⊢ ( ∃! 𝑦 ∈ 𝐴 𝜓 ↔ ( ∃ 𝑦 ∈ 𝐴 𝜓 ∧ ∃* 𝑦 ∈ 𝐴 𝜓 ) )
9	6 7 8	3bitr4i	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑦 ∈ 𝐴 𝜓 )