Step |
Hyp |
Ref |
Expression |
1 |
|
findes.1 |
|- [. (/) / x ]. ph |
2 |
|
findes.2 |
|- ( x e. _om -> ( ph -> [. suc x / x ]. ph ) ) |
3 |
|
dfsbcq2 |
|- ( z = (/) -> ( [ z / x ] ph <-> [. (/) / x ]. ph ) ) |
4 |
|
sbequ |
|- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) ) |
5 |
|
dfsbcq2 |
|- ( z = suc y -> ( [ z / x ] ph <-> [. suc y / x ]. ph ) ) |
6 |
|
sbequ12r |
|- ( z = x -> ( [ z / x ] ph <-> ph ) ) |
7 |
|
nfv |
|- F/ x y e. _om |
8 |
|
nfs1v |
|- F/ x [ y / x ] ph |
9 |
|
nfsbc1v |
|- F/ x [. suc y / x ]. ph |
10 |
8 9
|
nfim |
|- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph ) |
11 |
7 10
|
nfim |
|- F/ x ( y e. _om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) |
12 |
|
eleq1w |
|- ( x = y -> ( x e. _om <-> y e. _om ) ) |
13 |
|
sbequ12 |
|- ( x = y -> ( ph <-> [ y / x ] ph ) ) |
14 |
|
suceq |
|- ( x = y -> suc x = suc y ) |
15 |
14
|
sbceq1d |
|- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) ) |
16 |
13 15
|
imbi12d |
|- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) ) |
17 |
12 16
|
imbi12d |
|- ( x = y -> ( ( x e. _om -> ( ph -> [. suc x / x ]. ph ) ) <-> ( y e. _om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) ) ) |
18 |
11 17 2
|
chvarfv |
|- ( y e. _om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) |
19 |
3 4 5 6 1 18
|
finds |
|- ( x e. _om -> ph ) |