Theorem bj-cbvexvv 37275

Description: Existentially quantifying with respect to a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1911 and the existence axiom extru 1976. (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 1913	. . 3 ⊢ (∃𝑦𝜓 → 𝜓)
2		ax-5 1911	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3	1, 2	syl 17	. 2 ⊢ (∃𝑦𝜓 → ∀𝑥𝜓)
4		bj-axdd2 36792	. 2 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
5	3, 4	syl5 34	1 ⊢ (∃𝑥𝜑 → (∃𝑦𝜓 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by: (None)