Theorem bj-df-sb

Description: Proposed definition to replace df-sb and df-sbc . Proof is therefore unimportant. Contrary to df-sb , this definition makes a substituted formula false when one substitutes a non-existent object for a variable. This is better suited to the "Levy-style" treatment of classes as virtual objects adopted by set.mm. Indeed, in most set theories, and in particular in this database, one wants to be able to collect the extension of a formula, so it is natural that the extension of a formula contain only sets, and not proper classes. This is illustrated by the fact that bj-dfsbc holds unconditionally. That statement could therefore replace df-clab .

This definition is equivalent to df-sb when a setvar is substituted for the class, since, as soon as, ax6ev is posited, all variables "exist". (Contributed by BJ, 19-Feb-2026)

	Ref	Expression
Assertion	bj-df-sb	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists y\left(y=A\wedge \forall x\left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sbc7	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists y\left(y=A\wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)$$
2		sbsbc	$$⊢\left[y/x\right]\phi \leftrightarrow \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi$$
3		sb6	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
4	2 3	bitr3i	$$⊢\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
5	4	anbi2i	$$⊢\left(y=A\wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)\leftrightarrow \left(y=A\wedge \forall x\left(x=y\to \phi \right)\right)$$
6	5	exbii	$$⊢\exists y\left(y=A\wedge \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \right)\leftrightarrow \exists y\left(y=A\wedge \forall x\left(x=y\to \phi \right)\right)$$
7	1 6	bitri	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists y\left(y=A\wedge \forall x\left(x=y\to \phi \right)\right)$$