Theorem csbeq2dv

Description: Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005) (Revised by Mario Carneiro, 1-Sep-2015)

	Ref	Expression
Hypothesis	csbeq2dv.1	|- ( ph -> B = C )
Assertion	csbeq2dv	|- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )

Proof

Step	Hyp	Ref	Expression
1		csbeq2dv.1	|- ( ph -> B = C )
2	1	eleq2d	|- ( ph -> ( y e. B <-> y e. C ) )
3	2	sbcbidv	|- ( ph -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
4	3	abbidv	|- ( ph -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
5		df-csb	|- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
6		df-csb	|- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
7	4 5 6	3eqtr4g	|- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )