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Metamath Proof Explorer

Theorem cbvald

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . See cbvaldw for a version with x , y disjoint, not depending on ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Revised by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvald.1	$⊢ Ⅎ y φ$
2		cbvald.2	$⊢ φ \to Ⅎ y ψ$
3		cbvald.3	$⊢ φ \to (x = y \to (ψ \leftrightarrow χ))$
4		nfv	$⊢ Ⅎ x φ$
5		nfvd	$⊢ φ \to Ⅎ x χ$
6	4 1 2 5 3	cbv2	$⊢ φ \to (\forall x ψ \leftrightarrow \forall y χ)$