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Metamath Proof Explorer

Theorem cbv3

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbv3v if possible. (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv3.1	$⊢ Ⅎ y φ$
	cbv3.2	$⊢ Ⅎ x ψ$
	cbv3.3	$⊢ x = y \to (φ \to ψ)$
Assertion	cbv3	$⊢ \forall x φ \to \forall y ψ$

Proof

Step	Hyp	Ref	Expression
1		cbv3.1	$⊢ Ⅎ y φ$
2		cbv3.2	$⊢ Ⅎ x ψ$
3		cbv3.3	$⊢ x = y \to (φ \to ψ)$
4	1	nf5ri	$⊢ φ \to \forall y φ$
5	4	hbal	$⊢ \forall x φ \to \forall y \forall x φ$
6	2 3	spim	$⊢ \forall x φ \to ψ$
7	5 6	alrimih	$⊢ \forall x φ \to \forall y ψ$