Definition df-sbc

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

When A is a proper class, our definition evaluates to false (see sbcex ). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 for our definition, whose right-hand side 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 below). For example, if A is a proper class, Quine's substitution of A for y in 0 e. y evaluates to 0 e. 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 syntactic 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 definitiononly to prove theorem dfsbcq , which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc in the form of sbc8g . 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 and assert that [. A / x ]. ph is always false when A is a proper class.

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

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)

	Ref	Expression
Assertion	df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1		vx	⊢ 𝑥
2		wph	⊢ 𝜑
3	2 1 0	wsbc	⊢ [ 𝐴 / 𝑥 ] 𝜑
4	2 1	cab	⊢ { 𝑥 ∣ 𝜑 }
5	0 4	wcel	⊢ 𝐴 ∈ { 𝑥 ∣ 𝜑 }
6	3 5	wb	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } )