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	|- ( [. A / x ]. ph <-> E. y ( y = A /\ A. x ( x = y -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		sbc7	|- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )
2		sbsbc	|- ( [ y / x ] ph <-> [. y / x ]. ph )
3		sb6	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
4	2 3	bitr3i	|- ( [. y / x ]. ph <-> A. x ( x = y -> ph ) )
5	4	anbi2i	|- ( ( y = A /\ [. y / x ]. ph ) <-> ( y = A /\ A. x ( x = y -> ph ) ) )
6	5	exbii	|- ( E. y ( y = A /\ [. y / x ]. ph ) <-> E. y ( y = A /\ A. x ( x = y -> ph ) ) )
7	1 6	bitri	|- ( [. A / x ]. ph <-> E. y ( y = A /\ A. x ( x = y -> ph ) ) )