Theorem dfsbcq

Description: Proper substitution of a class for a set in a wff given equal classes. This is the essence of the sixth axiom of Frege, specifically Proposition 52 of Frege1879 p. 50.

This theorem, which is similar to Theorem 6.7 of Quine p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 instead of df-sbc . ( dfsbcq2 is needed because unlike Quine we do not overload the df-sb syntax.) As a consequence of these theorems, we can derive sbc8g , which is a weaker version of df-sbc that leaves substitution undefined when A is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g , so we will allow direct use of df-sbc after Theorem sbc2or below. Proper substitution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995)

	Ref	Expression
Assertion	dfsbcq	$$⊢A=B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		eleq1	$$⊢A=B\to \left(A\in \left\{x|\phi \right\}\leftrightarrow B\in \left\{x|\phi \right\}\right)$$
2		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow A\in \left\{x|\phi \right\}$$
3		df-sbc	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \leftrightarrow B\in \left\{x|\phi \right\}$$
4	1 2 3	3bitr4g	$$⊢A=B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$