Theorem cbval2

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbval2v if possible. (Contributed by NM, 22-Dec-2003) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-Sep-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval2.1	|- F/ z ph
	cbval2.2	|- F/ w ph
	cbval2.3	|- F/ x ps
	cbval2.4	|- F/ y ps
	cbval2.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
Assertion	cbval2	|- ( A. x A. y ph <-> A. z A. w ps )

Proof

Step	Hyp	Ref	Expression
1		cbval2.1	|- F/ z ph
2		cbval2.2	|- F/ w ph
3		cbval2.3	|- F/ x ps
4		cbval2.4	|- F/ y ps
5		cbval2.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
6	1	nfal	|- F/ z A. y ph
7	3	nfal	|- F/ x A. w ps
8		nfv	|- F/ y x = z
9		nfv	|- F/ w x = z
10	2	a1i	|- ( x = z -> F/ w ph )
11	4	a1i	|- ( x = z -> F/ y ps )
12	5	ex	|- ( x = z -> ( y = w -> ( ph <-> ps ) ) )
13	8 9 10 11 12	cbv2	|- ( x = z -> ( A. y ph <-> A. w ps ) )
14	6 7 13	cbval	|- ( A. x A. y ph <-> A. z A. w ps )