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Metamath Proof Explorer

Theorem bnj609

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 bnj609.1 
|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
bnj609.2 
|- ( ph" <-> [. G / f ]. ph )
bnj609.3 
|- G e. _V
Assertion bnj609 
|- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) )

Proof

Step Hyp Ref Expression
1 bnj609.1 
 |-  ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2 bnj609.2 
 |-  ( ph" <-> [. G / f ]. ph )
3 bnj609.3 
 |-  G e. _V
4 dfsbcq 
 |-  ( e = G -> ( [. e / f ]. ph <-> [. G / f ]. ph ) )
5 fveq1 
 |-  ( e = G -> ( e ` (/) ) = ( G ` (/) ) )
6 5 eqeq1d 
 |-  ( e = G -> ( ( e ` (/) ) = _pred ( X , A , R ) <-> ( G ` (/) ) = _pred ( X , A , R ) ) )
7 1 sbcbii 
 |-  ( [. e / f ]. ph <-> [. e / f ]. ( f ` (/) ) = _pred ( X , A , R ) )
8 vex 
 |-  e e. _V
9 fveq1 
 |-  ( f = e -> ( f ` (/) ) = ( e ` (/) ) )
10 9 eqeq1d 
 |-  ( f = e -> ( ( f ` (/) ) = _pred ( X , A , R ) <-> ( e ` (/) ) = _pred ( X , A , R ) ) )
11 8 10 sbcie 
 |-  ( [. e / f ]. ( f ` (/) ) = _pred ( X , A , R ) <-> ( e ` (/) ) = _pred ( X , A , R ) )
12 7 11 bitri 
 |-  ( [. e / f ]. ph <-> ( e ` (/) ) = _pred ( X , A , R ) )
13 3 4 6 12 vtoclb 
 |-  ( [. G / f ]. ph <-> ( G ` (/) ) = _pred ( X , A , R ) )
14 2 13 bitri 
 |-  ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) )