Theorem mojust

Description: Soundness justification theorem for df-mo (note that y and z need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010) Added this theorem by adapting the proof of eujust . (Revised by BJ, 30-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		equequ2	|- ( y = t -> ( x = y <-> x = t ) )
2	1	imbi2d	|- ( y = t -> ( ( ph -> x = y ) <-> ( ph -> x = t ) ) )
3	2	albidv	|- ( y = t -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = t ) ) )
4	3	cbvexvw	|- ( E. y A. x ( ph -> x = y ) <-> E. t A. x ( ph -> x = t ) )
5		equequ2	|- ( t = z -> ( x = t <-> x = z ) )
6	5	imbi2d	|- ( t = z -> ( ( ph -> x = t ) <-> ( ph -> x = z ) ) )
7	6	albidv	|- ( t = z -> ( A. x ( ph -> x = t ) <-> A. x ( ph -> x = z ) ) )
8	7	cbvexvw	|- ( E. t A. x ( ph -> x = t ) <-> E. z A. x ( ph -> x = z ) )
9	4 8	bitri	|- ( E. y A. x ( ph -> x = y ) <-> E. z A. x ( ph -> x = z ) )