Theorem csbeq2d

Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005) (Revised by Mario Carneiro, 1-Sep-2015)

	Ref	Expression
Hypotheses	csbeq2d.1	|- F/ x ph
	csbeq2d.2	|- ( ph -> B = C )
Assertion	csbeq2d	|- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )

Proof

Step	Hyp	Ref	Expression
1		csbeq2d.1	|- F/ x ph
2		csbeq2d.2	|- ( ph -> B = C )
3	2	eleq2d	|- ( ph -> ( y e. B <-> y e. C ) )
4	1 3	sbcbid	|- ( ph -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
5	4	abbidv	|- ( ph -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb	|- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb	|- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5 6 7	3eqtr4g	|- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )