Theorem bnj154

Description: Technical lemma for bnj153 . 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	bnj154.1	|- ( ph1 <-> [. g / f ]. ph' )
	bnj154.2	|- ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) )
Assertion	bnj154	|- ( ph1 <-> ( g ` (/) ) = _pred ( x , A , R ) )

Proof

Step	Hyp	Ref	Expression
1		bnj154.1	|- ( ph1 <-> [. g / f ]. ph' )
2		bnj154.2	|- ( ph' <-> ( f ` (/) ) = _pred ( x , A , R ) )
3	2	sbcbii	|- ( [. g / f ]. ph' <-> [. g / f ]. ( f ` (/) ) = _pred ( x , A , R ) )
4		vex	|- g e. _V
5		fveq1	|- ( f = g -> ( f ` (/) ) = ( g ` (/) ) )
6	5	eqeq1d	|- ( f = g -> ( ( f ` (/) ) = _pred ( x , A , R ) <-> ( g ` (/) ) = _pred ( x , A , R ) ) )
7	4 6	sbcie	|- ( [. g / f ]. ( f ` (/) ) = _pred ( x , A , R ) <-> ( g ` (/) ) = _pred ( x , A , R ) )
8	1 3 7	3bitri	|- ( ph1 <-> ( g ` (/) ) = _pred ( x , A , R ) )