Definition df-sbc 2855

Description: Define the proper substitution of a class for a set.

When A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 2879 for our definition, which always evaluates to true for proper classes.

Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 2856 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of ph, and it does not seem possible to express it with a single closed formula.

If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 2856, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2855 in the form of sbc8g 2861. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of A in every use of this definition) we allow direct reference to df-sbc 2855 and assert that [. A / x ]. ph is always false when A is a proper class.

The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)

Assertion

Ref	Expression
df-sbc	|- ( [. A / x ]. ph <-> A e. { x | ph } )

Detailed syntax breakdown of Definition df-sbc

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 setvar x
3		cA	. . 3 class A
4	1, 2, 3	wsbc 2854	. 2 wff [. A / x ]. ph
5	1, 2	cab 2081	. . 3 class { x | ph }
6	3, 5	wcel 1445	. 2 wff A e. { x | ph }
7	4, 6	wb 104	1 wff ( [. A / x ]. ph <-> A e. { x | ph } )

This definition is referenced by:  dfsbcq  2856  dfsbcq2  2857  sbcex  2862  nfsbc1d  2870  nfsbcd  2873  cbvsbc  2881  sbcbi2  2903  sbcbid  2910  intab  3739  brab1  3912  iotacl  5037  riotasbc  5661  bdsbcALT  12474