Theorem sbjust

Description: Justification theorem for df-sb proved from Tarski's FOL axiom schemes. (Contributed by BJ, 22-Jan-2023)

	Ref	Expression
Assertion	sbjust	|- ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. z ( z = t -> A. x ( x = z -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		equequ1	|- ( y = z -> ( y = t <-> z = t ) )
2		equequ2	|- ( y = z -> ( x = y <-> x = z ) )
3	2	imbi1d	|- ( y = z -> ( ( x = y -> ph ) <-> ( x = z -> ph ) ) )
4	3	albidv	|- ( y = z -> ( A. x ( x = y -> ph ) <-> A. x ( x = z -> ph ) ) )
5	1 4	imbi12d	|- ( y = z -> ( ( y = t -> A. x ( x = y -> ph ) ) <-> ( z = t -> A. x ( x = z -> ph ) ) ) )
6	5	cbvalvw	|- ( A. y ( y = t -> A. x ( x = y -> ph ) ) <-> A. z ( z = t -> A. x ( x = z -> ph ) ) )