Theorem cbvaldw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvald with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Gino Giotto, 10-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbvaldw.1	|- F/ y ph
2		cbvaldw.2	|- ( ph -> F/ y ps )
3		cbvaldw.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4		nfv	|- F/ x ph
5		nfvd	|- ( ph -> F/ x ch )
6	4 1 2 5 3	cbv2w	|- ( ph -> ( A. x ps <-> A. y ch ) )