Definition df-sbc 3048

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 3073 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 3049 below). For example, if A is a proper class, Quine's substitution of A for y in 0 ∈ y evaluates to 0 ∈ A rather than our falsehood. (This can be seen by substituting A, y, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactical breakdown of φ, 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 3049, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3048 in the form of sbc8g 3054. 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 3048 and assert that [̣A / x]̣φ is always false when A is a proper class.

Theorem sbc2or 3055 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3049.

The related definition df-csb 3138 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]̣φ ↔ A ∈ {x ∣ φ})

Detailed syntax breakdown of Definition df-sbc

Step	Hyp	Ref	Expression
1		wph	. . 3 wff φ
2		vx	. . 3 setvar x
3		cA	. . 3 class A
4	1, 2, 3	wsbc 3047	. 2 wff [̣A / x]̣φ
5	1, 2	cab 2339	. . 3 class {x ∣ φ}
6	3, 5	wcel 1710	. 2 wff A ∈ {x ∣ φ}
7	4, 6	wb 176	1 wff ([̣A / x]̣φ ↔ A ∈ {x ∣ φ})

This definition is referenced by:  dfsbcq  3049  dfsbcq2  3050  sbcex  3056  nfsbc1d  3064  nfsbcd  3067  cbvsbc  3075  sbcbid  3100  intab  3957  iotacl  4363  brab1  4685