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Metamath Proof Explorer

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	$⊢ \forall x φ \to \forall y \forall x φ$
	cbvaliw.2	$⊢ \neg ψ \to \forall x \neg ψ$
	cbvaliw.3	$⊢ x = y \to (φ \to ψ)$
Assertion	cbvaliw	$⊢ \forall x φ \to \forall y ψ$

Proof

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