Theorem bj-cbvexvv 37295

Description: Existentially quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1943 and the existence axiom extru 2008. See bj-cbvew 37297 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cbvexvv	⊢ (∃𝑥𝜑 → (∃𝑦𝜓 → ∃𝑥𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem bj-cbvexvv

Step	Hyp	Ref	Expression
1		ax5e 1945	. 2 ⊢ (∃𝑦𝜓 → 𝜓)
2		bj-spvew 37291	. . 3 ⊢ (∃𝑥𝜑 → (𝜓 ↔ ∃𝑥𝜓))
3	2	biimpd 232	. 2 ⊢ (∃𝑥𝜑 → (𝜓 → ∃𝑥𝜓))
4	1, 3	syl5 35	1 ⊢ (∃𝑥𝜑 → (∃𝑦𝜓 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-cbvew  37297