Theorem cbvaliw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of KalishMontague p. 86. (Contributed by NM, 19-Apr-2017)

	Ref	Expression
Hypotheses	cbvaliw.1	$$⊢\forall x\phi \to \forall y\forall x\phi$$
	cbvaliw.2	$$⊢\neg \psi \to \forall x\neg \psi$$
	cbvaliw.3	$$⊢x=y\to \left(\phi \to \psi \right)$$
Assertion	cbvaliw	$$⊢\forall x\phi \to \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvaliw.1	$$⊢\forall x\phi \to \forall y\forall x\phi$$
2		cbvaliw.2	$$⊢\neg \psi \to \forall x\neg \psi$$
3		cbvaliw.3	$$⊢x=y\to \left(\phi \to \psi \right)$$
4	2 3	spimw	$$⊢\forall x\phi \to \psi$$
5	1 4	alrimih	$$⊢\forall x\phi \to \forall y\psi$$