Theorem bnj581

Description: Technical lemma for bnj580 . 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) Remove unnecessary distinct variable conditions. (Revised by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj581.3	|- ( ch <-> ( f Fn n /\ ph /\ ps ) )
	bnj581.4	|- ( ph' <-> [. g / f ]. ph )
	bnj581.5	|- ( ps' <-> [. g / f ]. ps )
	bnj581.6	|- ( ch' <-> [. g / f ]. ch )
Assertion	bnj581	|- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )

Proof

Step	Hyp	Ref	Expression
1		bnj581.3	|- ( ch <-> ( f Fn n /\ ph /\ ps ) )
2		bnj581.4	|- ( ph' <-> [. g / f ]. ph )
3		bnj581.5	|- ( ps' <-> [. g / f ]. ps )
4		bnj581.6	|- ( ch' <-> [. g / f ]. ch )
5	1	sbcbii	|- ( [. g / f ]. ch <-> [. g / f ]. ( f Fn n /\ ph /\ ps ) )
6		sbc3an	|- ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> ( [. g / f ]. f Fn n /\ [. g / f ]. ph /\ [. g / f ]. ps ) )
7		bnj62	|- ( [. g / f ]. f Fn n <-> g Fn n )
8	7	bicomi	|- ( g Fn n <-> [. g / f ]. f Fn n )
9	8 2 3	3anbi123i	|- ( ( g Fn n /\ ph' /\ ps' ) <-> ( [. g / f ]. f Fn n /\ [. g / f ]. ph /\ [. g / f ]. ps ) )
10	6 9	bitr4i	|- ( [. g / f ]. ( f Fn n /\ ph /\ ps ) <-> ( g Fn n /\ ph' /\ ps' ) )
11	4 5 10	3bitri	|- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )