Theorem cbv1h

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) (Proof shortened by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv1h.1	|- ( ph -> ( ps -> A. y ps ) )
	cbv1h.2	|- ( ph -> ( ch -> A. x ch ) )
	cbv1h.3	|- ( ph -> ( x = y -> ( ps -> ch ) ) )
Assertion	cbv1h	|- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbv1h.1	|- ( ph -> ( ps -> A. y ps ) )
2		cbv1h.2	|- ( ph -> ( ch -> A. x ch ) )
3		cbv1h.3	|- ( ph -> ( x = y -> ( ps -> ch ) ) )
4		nfa1	|- F/ x A. x A. y ph
5		nfa2	|- F/ y A. x A. y ph
6		2sp	|- ( A. x A. y ph -> ph )
7	6 1	syl	|- ( A. x A. y ph -> ( ps -> A. y ps ) )
8	5 7	nf5d	|- ( A. x A. y ph -> F/ y ps )
9	6 2	syl	|- ( A. x A. y ph -> ( ch -> A. x ch ) )
10	4 9	nf5d	|- ( A. x A. y ph -> F/ x ch )
11	6 3	syl	|- ( A. x A. y ph -> ( x = y -> ( ps -> ch ) ) )
12	4 5 8 10 11	cbv1	|- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )