Theorem bj-cbvexiw

Description: Change bound variable. This is to cbvexvw what cbvaliw is to cbvalvw . TODO: move after cbvalivw . (Contributed by BJ, 17-Mar-2020)

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

Proof

Step	Hyp	Ref	Expression
1		bj-cbvexiw.1	⊢ ( ∃ 𝑥 ∃ 𝑦 𝜓 → ∃ 𝑦 𝜓 )
2		bj-cbvexiw.2	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		bj-cbvexiw.3	⊢ ( 𝑦 = 𝑥 → ( 𝜑 → 𝜓 ) )
4	2 3	spimew	⊢ ( 𝜑 → ∃ 𝑦 𝜓 )
5	1 4	bj-sylge	⊢ ( ∃ 𝑥 𝜑 → ∃ 𝑦 𝜓 )