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	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvalvw	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$

Proof

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