Theorem cbvald

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . See cbvaldw for a version with x , y disjoint, not depending on ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Revised by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvald.1	|- F/ y ph
2		cbvald.2	|- ( ph -> F/ y ps )
3		cbvald.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4		nfv	|- F/ x ph
5		nfvd	|- ( ph -> F/ x ch )
6	4 1 2 5 3	cbv2	|- ( ph -> ( A. x ps <-> A. y ch ) )