Definition df-csb 1998

Description: Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 1168, to prevent ambiguity. Theorem sbcel1g 2009 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2018 recreates substitution into a wff from this definition.

Assertion

Ref	Expression
df-csb	⊢ [A / x]B = {y∣[A / x]y ∈ B}

Distinct variable groups:   y,A   y,B   x,y

Detailed syntax breakdown of Definition df-csb

Step	Hyp	Ref	Expression
1		vx	. . 3 set x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	csb 1997	. 2 class [A / x]B
5		vy	. . . . . 6 set y
6	5	cv 953	. . . . 5 class y
7	6, 3	wcel 956	. . . 4 wff y ∈ B
8	7, 1, 2	wsbc 1168	. . 3 wff [A / x]y ∈ B
9	8, 5	cab 1461	. 2 class {y∣[A / x]y ∈ B}
10	4, 9	wceq 954	1 wff [A / x]B = {y∣[A / x]y ∈ B}

This definition is referenced by:  csbeq1 1999  csbid 2001  csbcog 2003  csbexg 2004  csbconstgf 2006  sbcel12g 2007  sbceqdig 2008  csbvarg 2017  hbcsb1g 2020  hbcsbg 2022  csbiegft 2025  csbabg 2039  fsump1f 6957

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