Step |
Hyp |
Ref |
Expression |
1 |
|
frege124d.f |
|- ( ph -> F e. _V ) |
2 |
|
frege124d.x |
|- ( ph -> X e. dom F ) |
3 |
|
frege124d.a |
|- ( ph -> A = ( F ` X ) ) |
4 |
|
frege124d.xb |
|- ( ph -> X ( t+ ` F ) B ) |
5 |
|
frege124d.fun |
|- ( ph -> Fun F ) |
6 |
3
|
eqcomd |
|- ( ph -> ( F ` X ) = A ) |
7 |
|
funbrfvb |
|- ( ( Fun F /\ X e. dom F ) -> ( ( F ` X ) = A <-> X F A ) ) |
8 |
5 2 7
|
syl2anc |
|- ( ph -> ( ( F ` X ) = A <-> X F A ) ) |
9 |
6 8
|
mpbid |
|- ( ph -> X F A ) |
10 |
|
funeu |
|- ( ( Fun F /\ X F A ) -> E! a X F a ) |
11 |
5 9 10
|
syl2anc |
|- ( ph -> E! a X F a ) |
12 |
|
fvex |
|- ( F ` X ) e. _V |
13 |
3 12
|
eqeltrdi |
|- ( ph -> A e. _V ) |
14 |
|
sbcan |
|- ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) <-> ( [. A / a ]. X F a /\ [. A / a ]. -. a ( t+ ` F ) B ) ) |
15 |
|
sbcbr2g |
|- ( A e. _V -> ( [. A / a ]. X F a <-> X F [_ A / a ]_ a ) ) |
16 |
|
csbvarg |
|- ( A e. _V -> [_ A / a ]_ a = A ) |
17 |
16
|
breq2d |
|- ( A e. _V -> ( X F [_ A / a ]_ a <-> X F A ) ) |
18 |
15 17
|
bitrd |
|- ( A e. _V -> ( [. A / a ]. X F a <-> X F A ) ) |
19 |
|
sbcng |
|- ( A e. _V -> ( [. A / a ]. -. a ( t+ ` F ) B <-> -. [. A / a ]. a ( t+ ` F ) B ) ) |
20 |
|
sbcbr1g |
|- ( A e. _V -> ( [. A / a ]. a ( t+ ` F ) B <-> [_ A / a ]_ a ( t+ ` F ) B ) ) |
21 |
16
|
breq1d |
|- ( A e. _V -> ( [_ A / a ]_ a ( t+ ` F ) B <-> A ( t+ ` F ) B ) ) |
22 |
20 21
|
bitrd |
|- ( A e. _V -> ( [. A / a ]. a ( t+ ` F ) B <-> A ( t+ ` F ) B ) ) |
23 |
22
|
notbid |
|- ( A e. _V -> ( -. [. A / a ]. a ( t+ ` F ) B <-> -. A ( t+ ` F ) B ) ) |
24 |
19 23
|
bitrd |
|- ( A e. _V -> ( [. A / a ]. -. a ( t+ ` F ) B <-> -. A ( t+ ` F ) B ) ) |
25 |
18 24
|
anbi12d |
|- ( A e. _V -> ( ( [. A / a ]. X F a /\ [. A / a ]. -. a ( t+ ` F ) B ) <-> ( X F A /\ -. A ( t+ ` F ) B ) ) ) |
26 |
14 25
|
syl5bb |
|- ( A e. _V -> ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) <-> ( X F A /\ -. A ( t+ ` F ) B ) ) ) |
27 |
13 26
|
syl |
|- ( ph -> ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) <-> ( X F A /\ -. A ( t+ ` F ) B ) ) ) |
28 |
|
spesbc |
|- ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) -> E. a ( X F a /\ -. a ( t+ ` F ) B ) ) |
29 |
27 28
|
syl6bir |
|- ( ph -> ( ( X F A /\ -. A ( t+ ` F ) B ) -> E. a ( X F a /\ -. a ( t+ ` F ) B ) ) ) |
30 |
9 29
|
mpand |
|- ( ph -> ( -. A ( t+ ` F ) B -> E. a ( X F a /\ -. a ( t+ ` F ) B ) ) ) |
31 |
|
eupicka |
|- ( ( E! a X F a /\ E. a ( X F a /\ -. a ( t+ ` F ) B ) ) -> A. a ( X F a -> -. a ( t+ ` F ) B ) ) |
32 |
11 30 31
|
syl6an |
|- ( ph -> ( -. A ( t+ ` F ) B -> A. a ( X F a -> -. a ( t+ ` F ) B ) ) ) |
33 |
|
alinexa |
|- ( A. a ( X F a -> -. a ( t+ ` F ) B ) <-> -. E. a ( X F a /\ a ( t+ ` F ) B ) ) |
34 |
|
funrel |
|- ( Fun F -> Rel F ) |
35 |
5 34
|
syl |
|- ( ph -> Rel F ) |
36 |
|
reltrclfv |
|- ( ( F e. _V /\ Rel F ) -> Rel ( t+ ` F ) ) |
37 |
1 35 36
|
syl2anc |
|- ( ph -> Rel ( t+ ` F ) ) |
38 |
|
brrelex2 |
|- ( ( Rel ( t+ ` F ) /\ X ( t+ ` F ) B ) -> B e. _V ) |
39 |
37 4 38
|
syl2anc |
|- ( ph -> B e. _V ) |
40 |
|
brcog |
|- ( ( X e. dom F /\ B e. _V ) -> ( X ( ( t+ ` F ) o. F ) B <-> E. a ( X F a /\ a ( t+ ` F ) B ) ) ) |
41 |
2 39 40
|
syl2anc |
|- ( ph -> ( X ( ( t+ ` F ) o. F ) B <-> E. a ( X F a /\ a ( t+ ` F ) B ) ) ) |
42 |
41
|
notbid |
|- ( ph -> ( -. X ( ( t+ ` F ) o. F ) B <-> -. E. a ( X F a /\ a ( t+ ` F ) B ) ) ) |
43 |
33 42
|
bitr4id |
|- ( ph -> ( A. a ( X F a -> -. a ( t+ ` F ) B ) <-> -. X ( ( t+ ` F ) o. F ) B ) ) |
44 |
32 43
|
sylibd |
|- ( ph -> ( -. A ( t+ ` F ) B -> -. X ( ( t+ ` F ) o. F ) B ) ) |
45 |
|
brdif |
|- ( X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B <-> ( X ( t+ ` F ) B /\ -. X ( ( t+ ` F ) o. F ) B ) ) |
46 |
45
|
simplbi2 |
|- ( X ( t+ ` F ) B -> ( -. X ( ( t+ ` F ) o. F ) B -> X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B ) ) |
47 |
4 44 46
|
sylsyld |
|- ( ph -> ( -. A ( t+ ` F ) B -> X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B ) ) |
48 |
|
trclfvdecomr |
|- ( F e. _V -> ( t+ ` F ) = ( F u. ( ( t+ ` F ) o. F ) ) ) |
49 |
1 48
|
syl |
|- ( ph -> ( t+ ` F ) = ( F u. ( ( t+ ` F ) o. F ) ) ) |
50 |
|
uncom |
|- ( F u. ( ( t+ ` F ) o. F ) ) = ( ( ( t+ ` F ) o. F ) u. F ) |
51 |
49 50
|
eqtrdi |
|- ( ph -> ( t+ ` F ) = ( ( ( t+ ` F ) o. F ) u. F ) ) |
52 |
|
eqimss |
|- ( ( t+ ` F ) = ( ( ( t+ ` F ) o. F ) u. F ) -> ( t+ ` F ) C_ ( ( ( t+ ` F ) o. F ) u. F ) ) |
53 |
51 52
|
syl |
|- ( ph -> ( t+ ` F ) C_ ( ( ( t+ ` F ) o. F ) u. F ) ) |
54 |
|
ssundif |
|- ( ( t+ ` F ) C_ ( ( ( t+ ` F ) o. F ) u. F ) <-> ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) C_ F ) |
55 |
53 54
|
sylib |
|- ( ph -> ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) C_ F ) |
56 |
55
|
ssbrd |
|- ( ph -> ( X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B -> X F B ) ) |
57 |
47 56
|
syld |
|- ( ph -> ( -. A ( t+ ` F ) B -> X F B ) ) |
58 |
|
funbrfv |
|- ( Fun F -> ( X F B -> ( F ` X ) = B ) ) |
59 |
5 57 58
|
sylsyld |
|- ( ph -> ( -. A ( t+ ` F ) B -> ( F ` X ) = B ) ) |
60 |
|
eqcom |
|- ( ( F ` X ) = B <-> B = ( F ` X ) ) |
61 |
59 60
|
syl6ib |
|- ( ph -> ( -. A ( t+ ` F ) B -> B = ( F ` X ) ) ) |
62 |
|
eqtr3 |
|- ( ( A = ( F ` X ) /\ B = ( F ` X ) ) -> A = B ) |
63 |
3 61 62
|
syl6an |
|- ( ph -> ( -. A ( t+ ` F ) B -> A = B ) ) |
64 |
63
|
orrd |
|- ( ph -> ( A ( t+ ` F ) B \/ A = B ) ) |