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Metamath Proof Explorer

Theorem cbval

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvalw , cbvalvw , cbvalv1 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993) (Revised by Mario Carneiro, 3-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbval.1	$⊢ Ⅎ y φ$
2		cbval.2	$⊢ Ⅎ x ψ$
3		cbval.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
4	3	biimpd	$⊢ x = y \to (φ \to ψ)$
5	1 2 4	cbv3	$⊢ \forall x φ \to \forall y ψ$
6	3	biimprd	$⊢ x = y \to (ψ \to φ)$
7	6	equcoms	$⊢ y = x \to (ψ \to φ)$
8	2 1 7	cbv3	$⊢ \forall y ψ \to \forall x φ$
9	5 8	impbii	$⊢ \forall x φ \leftrightarrow \forall y ψ$