Theorem cbvsbc

Description: Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvsbcw when possible. (Contributed by Jeff Hankins, 19-Sep-2009) (Proof shortened by Andrew Salmon, 8-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvsbc.1	|- F/ y ph
	cbvsbc.2	|- F/ x ps
	cbvsbc.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvsbc	|- ( [. A / x ]. ph <-> [. A / y ]. ps )

Proof

Step	Hyp	Ref	Expression
1		cbvsbc.1	|- F/ y ph
2		cbvsbc.2	|- F/ x ps
3		cbvsbc.3	|- ( x = y -> ( ph <-> ps ) )
4	1 2 3	cbvab	|- { x | ph } = { y | ps }
5	4	eleq2i	|- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc	|- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc	|- ( [. A / y ]. ps <-> A e. { y | ps } )
8	5 6 7	3bitr4i	|- ( [. A / x ]. ph <-> [. A / y ]. ps )