Theorem bj-cbvaw 37304

Description: Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37302. If ⊥ is substituted for 𝜑, then the statement reads: "universally quantifying over a non-occurring variable is independent from the variable as soon as that result is true for the False truth constant". The label "cbvaw" means "'change bound variable' theorem, 'all' quantifier, weak version". (Contributed by BJ, 14-Mar-2026.) This proof is not intuitionistic (it uses ja 188); an intuitionistically valid statement is obtained by expressing the antecedent as a disjunction (classically equivalent through imor 867). (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cbvaw	⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∀𝑥𝜓 → ∀𝑦𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem bj-cbvaw

Step	Hyp	Ref	Expression
1		exnal 1860	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
2		bj-cbvalvv 37302	. . 3 ⊢ (∃𝑥 ¬ 𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))
3	1, 2	sylbir 238	. 2 ⊢ (¬ ∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))
4		falim 1587	. . . 4 ⊢ (⊥ → 𝜓)
5	4	alimi 1844	. . 3 ⊢ (∀𝑦⊥ → ∀𝑦𝜓)
6	5	a1d 26	. 2 ⊢ (∀𝑦⊥ → (∀𝑥𝜓 → ∀𝑦𝜓))
7	3, 6	ja 188	1 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∀𝑥𝜓 → ∀𝑦𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ⊥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)