Theorem cbvrexvw

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv with a disjoint variable condition, which does not require ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 2-Jun-1998) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypothesis	cbvralvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvrexvw	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvralvw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
3	2 1	anbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) ) )
4	3	cbvexvw	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) )
5		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
6		df-rex	⊢ ( ∃ 𝑦 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜓 ) )
7	4 5 6	3bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 𝜓 )