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 ) )