Step |
Hyp |
Ref |
Expression |
1 |
|
bnj207.1 |
|- ( ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) ) |
2 |
|
bnj207.2 |
|- ( ph' <-> [. M / n ]. ph ) |
3 |
|
bnj207.3 |
|- ( ps' <-> [. M / n ]. ps ) |
4 |
|
bnj207.4 |
|- ( ch' <-> [. M / n ]. ch ) |
5 |
|
bnj207.5 |
|- M e. _V |
6 |
1
|
sbcbii |
|- ( [. M / n ]. ch <-> [. M / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) ) |
7 |
|
nfv |
|- F/ n ( R _FrSe A /\ x e. A ) |
8 |
7
|
sbc19.21g |
|- ( M e. _V -> ( [. M / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) ) ) ) |
9 |
5 8
|
ax-mp |
|- ( [. M / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) ) ) |
10 |
5
|
bnj89 |
|- ( [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) <-> E! f [. M / n ]. ( f Fn n /\ ph /\ ps ) ) |
11 |
5
|
bnj90 |
|- ( [. M / n ]. f Fn n <-> f Fn M ) |
12 |
11
|
bicomi |
|- ( f Fn M <-> [. M / n ]. f Fn n ) |
13 |
12 2 3 5
|
bnj206 |
|- ( [. M / n ]. ( f Fn n /\ ph /\ ps ) <-> ( f Fn M /\ ph' /\ ps' ) ) |
14 |
13
|
eubii |
|- ( E! f [. M / n ]. ( f Fn n /\ ph /\ ps ) <-> E! f ( f Fn M /\ ph' /\ ps' ) ) |
15 |
10 14
|
bitri |
|- ( [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) <-> E! f ( f Fn M /\ ph' /\ ps' ) ) |
16 |
15
|
imbi2i |
|- ( ( ( R _FrSe A /\ x e. A ) -> [. M / n ]. E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) ) |
17 |
9 16
|
bitri |
|- ( [. M / n ]. ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) ) |
18 |
6 17
|
bitri |
|- ( [. M / n ]. ch <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) ) |
19 |
4 18
|
bitri |
|- ( ch' <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn M /\ ph' /\ ps' ) ) ) |