Theorem cbveu

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbveuw , cbveuvw when possible. (Contributed by NM, 25-Nov-1994) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbveu.1	|- F/ y ph
	cbveu.2	|- F/ x ps
	cbveu.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbveu	|- ( E! x ph <-> E! y ps )

Proof

Step	Hyp	Ref	Expression
1		cbveu.1	|- F/ y ph
2		cbveu.2	|- F/ x ps
3		cbveu.3	|- ( x = y -> ( ph <-> ps ) )
4	1	sb8eu	|- ( E! x ph <-> E! y [ y / x ] ph )
5	2 3	sbie	|- ( [ y / x ] ph <-> ps )
6	5	eubii	|- ( E! y [ y / x ] ph <-> E! y ps )
7	4 6	bitri	|- ( E! x ph <-> E! y ps )