Theorem bj-cbval

Description: Changing a bound variable (universal quantification case) in a weak axiomatization that assumes that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023) Proved from ax-1 -- ax-5 . (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.equcomiv	$$⊢y=x\to x=y$$
	bj-cbval.nf0	$$⊢\phi \to \forall x\phi$$
	bj-cbval.nf1	$$⊢\phi \to \forall y\phi$$
	bj-cbval.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bj-cbval	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

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.equcomiv	$$⊢y=x\to x=y$$
4		bj-cbval.nf0	$$⊢\phi \to \forall x\phi$$
5		bj-cbval.nf1	$$⊢\phi \to \forall y\phi$$
6		bj-cbval.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
7		ax5e	$$⊢\exists x\chi \to \chi$$
8	7	a1i	$$⊢\phi \to \left(\exists x\chi \to \chi \right)$$
9	1	a1i	$$⊢\phi \to \forall y\exists xx=y$$
10	6	biimpd	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \to \chi \right)$$
11	4 5 8 9 10	bj-cbvalimdv	$$⊢\phi \to \left(\forall x\psi \to \forall y\chi \right)$$
12		ax5e	$$⊢\exists y\psi \to \psi$$
13	12	a1i	$$⊢\phi \to \left(\exists y\psi \to \psi \right)$$
14	2	a1i	$$⊢\phi \to \forall x\exists yy=x$$
15	6	biimprd	$$⊢\left(\phi \wedge x=y\right)\to \left(\chi \to \psi \right)$$
16	3 15	sylan2	$$⊢\left(\phi \wedge y=x\right)\to \left(\chi \to \psi \right)$$
17	5 4 13 14 16	bj-cbvalimdv	$$⊢\phi \to \left(\forall y\chi \to \forall x\psi \right)$$
18	11 17	impbid	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$