Theorem bj-cbval

Description: Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbval.denote	$$⊢\forall y\exists xx=y$$
	bj-cbval.denote2	$$⊢\forall x\exists yy=x$$
	bj-cbval.maj	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
	bj-cbval.equcomiv	$$⊢y=x\to x=y$$
Assertion	bj-cbval	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbval.denote	$$⊢\forall y\exists xx=y$$
2		bj-cbval.denote2	$$⊢\forall x\exists yy=x$$
3		bj-cbval.maj	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4		bj-cbval.equcomiv	$$⊢y=x\to x=y$$
5	3	biimpd	$$⊢x=y\to \left(\phi \to \psi \right)$$
6	5 1	bj-cbvalimi	$$⊢\forall x\phi \to \forall y\psi$$
7	3	biimprd	$$⊢x=y\to \left(\psi \to \phi \right)$$
8	4 7	syl	$$⊢y=x\to \left(\psi \to \phi \right)$$
9	8 2	bj-cbvalimi	$$⊢\forall y\psi \to \forall x\phi$$
10	6 9	impbii	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$