Theorem bj-cbvaw

Description: Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv . If F. is substituted for ph , 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 ); an intuitionistically valid statement is obtained by expressing the antecedent as a disjunction (classically equivalent through imor ). (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-cbvaw	$$⊢\left(\forall x\phi \to \forall y\perp \right)\to \left(\forall x\psi \to \forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		exnal	$$⊢\exists x\neg \phi \leftrightarrow \neg \forall x\phi$$
2		bj-cbvalvv	$$⊢\exists x\neg \phi \to \left(\forall x\psi \to \forall y\psi \right)$$
3	1 2	sylbir	$$⊢\neg \forall x\phi \to \left(\forall x\psi \to \forall y\psi \right)$$
4		falim	$$⊢\perp \to \psi$$
5	4	alimi	$$⊢\forall y\perp \to \forall y\psi$$
6	5	a1d	$$⊢\forall y\perp \to \left(\forall x\psi \to \forall y\psi \right)$$
7	3 6	ja	$$⊢\left(\forall x\phi \to \forall y\perp \right)\to \left(\forall x\psi \to \forall y\psi \right)$$