Theorem bj-cbvalvv 37630

Description: Universally quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1942 and the existence axiom extru 2007. See bj-cbvaw 37632 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem bj-cbvalvv

Step	Hyp	Ref	Expression
1		bj-spvw 37626	. . 3 ⊢ (∃𝑥𝜑 → (𝜓 ↔ ∀𝑥𝜓))
2	1	biimprd 251	. 2 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → 𝜓))
3		ax-5 1942	. 2 ⊢ (𝜓 → ∀𝑦𝜓)
4	2, 3	syl6 36	1 ⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  bj-cbvaw  37632  bj-cbveaw  37634