Theorem bj-cbvaw 36988

Description: Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 36986. 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 187); an intuitionistically valid statement is obtained by expressing the antecedent as a disjunction (classically equivalent through imor 859). (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 1834	. . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
2		bj-cbvalvv 36986	. . 3 ⊢ (∃𝑥 ¬ 𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))
3	1, 2	sylbir 236	. 2 ⊢ (¬ ∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))
4		falim 1564	. . . 4 ⊢ (⊥ → 𝜓)
5	4	alimi 1818	. . 3 ⊢ (∀𝑦⊥ → ∀𝑦𝜓)
6	5	a1d 25	. 2 ⊢ (∀𝑦⊥ → (∀𝑥𝜓 → ∀𝑦𝜓))
7	3, 6	ja 187	1 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∀𝑥𝜓 → ∀𝑦𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1545  ⊥wfal 1559  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917

This theorem depends on definitions:  df-bi 208  df-tru 1550  df-fal 1560  df-ex 1787

This theorem is referenced by: (None)