Theorem bnj121

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj121.1	|- ( ze <-> ( ( R _FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
	bnj121.2	|- ( ze' <-> [. 1o / n ]. ze )
	bnj121.3	|- ( ph' <-> [. 1o / n ]. ph )
	bnj121.4	|- ( ps' <-> [. 1o / n ]. ps )
Assertion	bnj121	|- ( ze' <-> ( ( R _FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj121.1	|- ( ze <-> ( ( R _FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
2		bnj121.2	|- ( ze' <-> [. 1o / n ]. ze )
3		bnj121.3	|- ( ph' <-> [. 1o / n ]. ph )
4		bnj121.4	|- ( ps' <-> [. 1o / n ]. ps )
5	1	sbcbii	|- ( [. 1o / n ]. ze <-> [. 1o / n ]. ( ( R _FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
6		bnj105	|- 1o e. _V
7	6	bnj90	|- ( [. 1o / n ]. f Fn n <-> f Fn 1o )
8	7	bicomi	|- ( f Fn 1o <-> [. 1o / n ]. f Fn n )
9	8 3 4	3anbi123i	|- ( ( f Fn 1o /\ ph' /\ ps' ) <-> ( [. 1o / n ]. f Fn n /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) )
10		sbc3an	|- ( [. 1o / n ]. ( f Fn n /\ ph /\ ps ) <-> ( [. 1o / n ]. f Fn n /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) )
11	9 10	bitr4i	|- ( ( f Fn 1o /\ ph' /\ ps' ) <-> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) )
12	11	imbi2i	|- ( ( ( R _FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) <-> ( ( R _FrSe A /\ x e. A ) -> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) )
13		nfv	|- F/ n ( R _FrSe A /\ x e. A )
14	13	sbc19.21g	|- ( 1o e. _V -> ( [. 1o / n ]. ( ( R _FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) ) )
15	6 14	ax-mp	|- ( [. 1o / n ]. ( ( R _FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> [. 1o / n ]. ( f Fn n /\ ph /\ ps ) ) )
16	12 15	bitr4i	|- ( ( ( R _FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) <-> [. 1o / n ]. ( ( R _FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
17	5 2 16	3bitr4i	|- ( ze' <-> ( ( R _FrSe A /\ x e. A ) -> ( f Fn 1o /\ ph' /\ ps' ) ) )