Theorem bj-cbvexw

Description: Change bound variable. This is to cbvexvw what cbvalw is to cbvalvw . (Contributed by BJ, 17-Mar-2020)

	Ref	Expression
Hypotheses	bj-cbvexw.1	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 )
	bj-cbvexw.2	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	bj-cbvexw.3	⊢ ( ∃ 𝑦 ∃ 𝑥 𝜑 → ∃ 𝑥 𝜑 )
	bj-cbvexw.4	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
	bj-cbvexw.5	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	bj-cbvexw	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-cbvexw.1	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 )
2		bj-cbvexw.2	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		bj-cbvexw.3	⊢ ( ∃ 𝑦 ∃ 𝑥 𝜑 → ∃ 𝑥 𝜑 )
4		bj-cbvexw.4	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
5		bj-cbvexw.5	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
6	5	equcoms	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
7	6	biimpd	⊢ ( 𝑦 = 𝑥 → ( 𝜑 → 𝜓 ) )
8	1 2 7	bj-cbvexiw	⊢ ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 )
9	5	biimprd	⊢ ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) )
10	3 4 9	bj-cbvexiw	⊢ ( ∃ 𝑦 𝜓 → ∃ 𝑥 𝜑 )
11	8 10	impbii	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 )