Theorem cbvrex2vw

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v with a disjoint variable condition, which does not require ax-13 . (Contributed by FL, 2-Jul-2012) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvrex2vw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜒 ) )
	cbvrex2vw.2	⊢ ( 𝑦 = 𝑤 → ( 𝜒 ↔ 𝜓 ) )
Assertion	cbvrex2vw	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvrex2vw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜒 ) )
2		cbvrex2vw.2	⊢ ( 𝑦 = 𝑤 → ( 𝜒 ↔ 𝜓 ) )
3	1	rexbidv	⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) )
4	3	cbvrexvw	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 )
5	2	cbvrexvw	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜒 ↔ ∃ 𝑤 ∈ 𝐵 𝜓 )
6	5	rexbii	⊢ ( ∃ 𝑧 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝜓 )
7	4 6	bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝜓 )