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Metamath Proof Explorer

Theorem bnj526

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 bnj526.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj526.2 
|- ( ph" <-> [. G / f ]. ph )
bnj526.3 
|- G e. _V
Assertion bnj526 
|- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) )

Proof

Step Hyp Ref Expression
1 bnj526.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj526.2 
 |-  ( ph" <-> [. G / f ]. ph )
3 bnj526.3 
 |-  G e. _V
4 1 sbcbii 
 |-  ( [. G / f ]. ph <-> [. G / f ]. ( f ` (/) ) = _pred ( X , A , R ) )
5 fveq1 
 |-  ( f = G -> ( f ` (/) ) = ( G ` (/) ) )
6 5 eqeq1d 
 |-  ( f = G -> ( ( f ` (/) ) = _pred ( X , A , R ) <-> ( G ` (/) ) = _pred ( X , A , R ) ) )
7 3 6 sbcie 
 |-  ( [. G / f ]. ( f ` (/) ) = _pred ( X , A , R ) <-> ( G ` (/) ) = _pred ( X , A , R ) )
8 2 4 7 3bitri 
 |-  ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) )