Metamath Proof Explorer


Theorem bnj523

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj523.1
|- ( ph <-> ( F ` (/) ) = _pred ( X , A , R ) )
bnj523.2
|- ( ph' <-> [. M / n ]. ph )
bnj523.3
|- M e. _V
Assertion bnj523
|- ( ph' <-> ( F ` (/) ) = _pred ( X , A , R ) )

Proof

Step Hyp Ref Expression
1 bnj523.1
 |-  ( ph <-> ( F ` (/) ) = _pred ( X , A , R ) )
2 bnj523.2
 |-  ( ph' <-> [. M / n ]. ph )
3 bnj523.3
 |-  M e. _V
4 1 sbcbii
 |-  ( [. M / n ]. ph <-> [. M / n ]. ( F ` (/) ) = _pred ( X , A , R ) )
5 3 bnj525
 |-  ( [. M / n ]. ( F ` (/) ) = _pred ( X , A , R ) <-> ( F ` (/) ) = _pred ( X , A , R ) )
6 2 4 5 3bitri
 |-  ( ph' <-> ( F ` (/) ) = _pred ( X , A , R ) )