Theorem bnj934

Description: Technical lemma for bnj69 . 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	bnj934.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
	bnj934.4	|- ( ph' <-> [. p / n ]. ph )
	bnj934.7	|- ( ph" <-> [. G / f ]. ph' )
	bnj934.50	|- G e. _V
Assertion	bnj934	|- ( ( ph /\ ( G ` (/) ) = ( f ` (/) ) ) -> ph" )

Proof

Step	Hyp	Ref	Expression
1		bnj934.1	|- ( ph <-> ( f ` (/) ) = _pred ( X , A , R ) )
2		bnj934.4	|- ( ph' <-> [. p / n ]. ph )
3		bnj934.7	|- ( ph" <-> [. G / f ]. ph' )
4		bnj934.50	|- G e. _V
5		eqtr	|- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ( f ` (/) ) = _pred ( X , A , R ) ) -> ( G ` (/) ) = _pred ( X , A , R ) )
6	1 5	sylan2b	|- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ph ) -> ( G ` (/) ) = _pred ( X , A , R ) )
7		vex	|- p e. _V
8	1 2 7	bnj523	|- ( ph' <-> ( f ` (/) ) = _pred ( X , A , R ) )
9	8 1	bitr4i	|- ( ph' <-> ph )
10	9	sbcbii	|- ( [. G / f ]. ph' <-> [. G / f ]. ph )
11	3 10	bitri	|- ( ph" <-> [. G / f ]. ph )
12	1 11 4	bnj609	|- ( ph" <-> ( G ` (/) ) = _pred ( X , A , R ) )
13	6 12	sylibr	|- ( ( ( G ` (/) ) = ( f ` (/) ) /\ ph ) -> ph" )
14	13	ancoms	|- ( ( ph /\ ( G ` (/) ) = ( f ` (/) ) ) -> ph" )