Theorem cbv2w

Description: Rule used to change bound variables, using implicit substitution. Version of cbv2 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbv2w.1	|- F/ x ph
	cbv2w.2	|- F/ y ph
	cbv2w.3	|- ( ph -> F/ y ps )
	cbv2w.4	|- ( ph -> F/ x ch )
	cbv2w.5	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
Assertion	cbv2w	|- ( ph -> ( A. x ps <-> A. y ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbv2w.1	|- F/ x ph
2		cbv2w.2	|- F/ y ph
3		cbv2w.3	|- ( ph -> F/ y ps )
4		cbv2w.4	|- ( ph -> F/ x ch )
5		cbv2w.5	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6		biimp	|- ( ( ps <-> ch ) -> ( ps -> ch ) )
7	5 6	syl6	|- ( ph -> ( x = y -> ( ps -> ch ) ) )
8	1 2 3 4 7	cbv1v	|- ( ph -> ( A. x ps -> A. y ch ) )
9		equcomi	|- ( y = x -> x = y )
10		biimpr	|- ( ( ps <-> ch ) -> ( ch -> ps ) )
11	9 5 10	syl56	|- ( ph -> ( y = x -> ( ch -> ps ) ) )
12	2 1 4 3 11	cbv1v	|- ( ph -> ( A. y ch -> A. x ps ) )
13	8 12	impbid	|- ( ph -> ( A. x ps <-> A. y ch ) )