Theorem bj-cbveaw 37375

Description: Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37371. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cbveaw	⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∀𝑦𝜓 → ∀𝑥𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem bj-cbveaw

Step	Hyp	Ref	Expression
1		empty 1939	. . 3 ⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
2		falim 1587	. . . . 5 ⊢ (⊥ → 𝜓)
3	2	alimi 1844	. . . 4 ⊢ (∀𝑥⊥ → ∀𝑥𝜓)
4	3	a1d 26	. . 3 ⊢ (∀𝑥⊥ → (∀𝑦𝜓 → ∀𝑥𝜓))
5	1, 4	sylbi 220	. 2 ⊢ (¬ ∃𝑥⊤ → (∀𝑦𝜓 → ∀𝑥𝜓))
6		bj-cbvalvv 37371	. 2 ⊢ (∃𝑦𝜑 → (∀𝑦𝜓 → ∀𝑥𝜓))
7	5, 6	ja 188	1 ⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∀𝑦𝜓 → ∀𝑥𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ⊤wtru 1571  ⊥wfal 1582  ∃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-tru 1573  df-fal 1583  df-ex 1813

This theorem is used by: (None)