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Metamath Proof Explorer

Theorem cbvalvw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvalv for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Apr-2017) (Proof shortened by Wolf Lammen, 28-Feb-2018)

		Ref	Expression
	Hypothesis	cbvalvw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
	Assertion	cbvalvw	$⊢ \forall x φ \leftrightarrow \forall y ψ$

Proof

Step	Hyp	Ref	Expression
1		cbvalvw.1	$⊢ x = y \to (φ \leftrightarrow ψ)$
2		ax-5	$⊢ \forall x φ \to \forall y \forall x φ$
3		ax-5	$⊢ \neg ψ \to \forall x \neg ψ$
4		ax-5	$⊢ \forall y ψ \to \forall x \forall y ψ$
5		ax-5	$⊢ \neg φ \to \forall y \neg φ$
6	2 3 4 5 1	cbvalw	$⊢ \forall x φ \leftrightarrow \forall y ψ$