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 ) )