Theorem dfsb

Description: Simplify definition df-sb by proving the renaming independency. (Contributed by Wolf Lammen, 5-Feb-2026) df-sb changed. (Revised by Wolf Lammen, 4-Jun-2026)

	Ref	Expression
Assertion	dfsb	|- ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfsbimp	|- ( [ t / x ] ph -> A. y ( y = t -> A. x ( x = y -> ph ) ) )
2		df-sb	|- ( [ t / x ] ph <-> ( A. y ( y = t -> A. x ( x = y -> ph ) ) /\ A. z ( z = t -> A. x ( x = z -> ph ) ) ) )
3		rename-sb	|- ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. z ( z = t -> A. x ( x = z -> ph ) ) )
4	2 3	just3-df	|- ( A. y ( y = t -> A. x ( x = y -> ph ) ) -> [ t / x ] ph )
5	1 4	impbii	|- ( [ t / x ] ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )