Theorem bj-cbv2hv

Description: Version of cbv2h with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbv2hv.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
	bj-cbv2hv.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
	bj-cbv2hv.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	bj-cbv2hv	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-cbv2hv.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
2		bj-cbv2hv.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
3		bj-cbv2hv.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
4		biimp	$$⊢\left(\psi \leftrightarrow \chi \right)\to \left(\psi \to \chi \right)$$
5	3 4	syl6	$$⊢\phi \to \left(x=y\to \left(\psi \to \chi \right)\right)$$
6	1 2 5	bj-cbv1hv	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \to \forall y\chi \right)$$
7		equcomi	$$⊢y=x\to x=y$$
8		biimpr	$$⊢\left(\psi \leftrightarrow \chi \right)\to \left(\chi \to \psi \right)$$
9	7 3 8	syl56	$$⊢\phi \to \left(y=x\to \left(\chi \to \psi \right)\right)$$
10	2 1 9	bj-cbv1hv	$$⊢\forall y\forall x\phi \to \left(\forall y\chi \to \forall x\psi \right)$$
11	10	alcoms	$$⊢\forall x\forall y\phi \to \left(\forall y\chi \to \forall x\psi \right)$$
12	6 11	impbid	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$