Theorem cbv1v

Description: Rule used to change bound variables, using implicit substitution. Version of cbv1 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 16-Jun-2019)

	Ref	Expression
Hypotheses	cbv1v.1	|- F/ x ph
	cbv1v.2	|- F/ y ph
	cbv1v.3	|- ( ph -> F/ y ps )
	cbv1v.4	|- ( ph -> F/ x ch )
	cbv1v.5	|- ( ph -> ( x = y -> ( ps -> ch ) ) )
Assertion	cbv1v	|- ( ph -> ( A. x ps -> A. y ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbv1v.1	|- F/ x ph
2		cbv1v.2	|- F/ y ph
3		cbv1v.3	|- ( ph -> F/ y ps )
4		cbv1v.4	|- ( ph -> F/ x ch )
5		cbv1v.5	|- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	2 3	nfim1	|- F/ y ( ph -> ps )
7	1 4	nfim1	|- F/ x ( ph -> ch )
8	5	com12	|- ( x = y -> ( ph -> ( ps -> ch ) ) )
9	8	a2d	|- ( x = y -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
10	6 7 9	cbv3v	|- ( A. x ( ph -> ps ) -> A. y ( ph -> ch ) )
11	1	19.21	|- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
12	2	19.21	|- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) )
13	10 11 12	3imtr3i	|- ( ( ph -> A. x ps ) -> ( ph -> A. y ch ) )
14	13	pm2.86i	|- ( ph -> ( A. x ps -> A. y ch ) )