This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )