Theorem qseq2

Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995)

	Ref	Expression
Assertion	qseq2	|- ( A = B -> ( C /. A ) = ( C /. B ) )

Proof

Step	Hyp	Ref	Expression
1		eceq2	|- ( A = B -> [ x ] A = [ x ] B )
2	1	eqeq2d	|- ( A = B -> ( y = [ x ] A <-> y = [ x ] B ) )
3	2	rexbidv	|- ( A = B -> ( E. x e. C y = [ x ] A <-> E. x e. C y = [ x ] B ) )
4	3	abbidv	|- ( A = B -> { y | E. x e. C y = [ x ] A } = { y | E. x e. C y = [ x ] B } )
5		df-qs	|- ( C /. A ) = { y | E. x e. C y = [ x ] A }
6		df-qs	|- ( C /. B ) = { y | E. x e. C y = [ x ] B }
7	4 5 6	3eqtr4g	|- ( A = B -> ( C /. A ) = ( C /. B ) )