Theorem cbv2h

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 11-May-1993) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv2h.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
	cbv2h.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
	cbv2h.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	cbv2h	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbv2h.1	$$⊢\phi \to \left(\psi \to \forall y\psi \right)$$
2		cbv2h.2	$$⊢\phi \to \left(\chi \to \forall x\chi \right)$$
3		cbv2h.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	cbv1h	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \to \forall y\chi \right)$$
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	cbv1h	$$⊢\forall y\forall x\phi \to \left(\forall y\chi \to \forall x\psi \right)$$
11	10	alcoms	$$⊢\forall x\forall y\phi \to \left(\forall y\chi \to \forall x\psi \right)$$
12	6 11	impbid	$$⊢\forall x\forall y\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$