Theorem cbval

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvalw , cbvalvw , cbvalv1 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993) (Revised by Mario Carneiro, 3-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval.1	|- F/ y ph
	cbval.2	|- F/ x ps
	cbval.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbval	|- ( A. x ph <-> A. y ps )

Proof

Step	Hyp	Ref	Expression
1		cbval.1	|- F/ y ph
2		cbval.2	|- F/ x ps
3		cbval.3	|- ( x = y -> ( ph <-> ps ) )
4	3	biimpd	|- ( x = y -> ( ph -> ps ) )
5	1 2 4	cbv3	|- ( A. x ph -> A. y ps )
6	3	biimprd	|- ( x = y -> ( ps -> ph ) )
7	6	equcoms	|- ( y = x -> ( ps -> ph ) )
8	2 1 7	cbv3	|- ( A. y ps -> A. x ph )
9	5 8	impbii	|- ( A. x ph <-> A. y ps )