Theorem cbv1

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbv1v with disjoint variable conditions, not depending on ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 3-Oct-2016) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv1.1	|- F/ x ph
	cbv1.2	|- F/ y ph
	cbv1.3	|- ( ph -> F/ y ps )
	cbv1.4	|- ( ph -> F/ x ch )
	cbv1.5	|- ( ph -> ( x = y -> ( ps -> ch ) ) )
Assertion	cbv1	|- ( ph -> ( A. x ps -> A. y ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbv1.1	|- F/ x ph
2		cbv1.2	|- F/ y ph
3		cbv1.3	|- ( ph -> F/ y ps )
4		cbv1.4	|- ( ph -> F/ x ch )
5		cbv1.5	|- ( 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	cbv3	|- ( 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 ) )