Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
|- ( x = 1 -> ( x + 1 ) = ( 1 + 1 ) ) |
2 |
|
oveq2 |
|- ( x = 1 -> ( 1 + x ) = ( 1 + 1 ) ) |
3 |
1 2
|
eqeq12d |
|- ( x = 1 -> ( ( x + 1 ) = ( 1 + x ) <-> ( 1 + 1 ) = ( 1 + 1 ) ) ) |
4 |
|
oveq1 |
|- ( x = y -> ( x + 1 ) = ( y + 1 ) ) |
5 |
|
oveq2 |
|- ( x = y -> ( 1 + x ) = ( 1 + y ) ) |
6 |
4 5
|
eqeq12d |
|- ( x = y -> ( ( x + 1 ) = ( 1 + x ) <-> ( y + 1 ) = ( 1 + y ) ) ) |
7 |
|
oveq1 |
|- ( x = ( y + 1 ) -> ( x + 1 ) = ( ( y + 1 ) + 1 ) ) |
8 |
|
oveq2 |
|- ( x = ( y + 1 ) -> ( 1 + x ) = ( 1 + ( y + 1 ) ) ) |
9 |
7 8
|
eqeq12d |
|- ( x = ( y + 1 ) -> ( ( x + 1 ) = ( 1 + x ) <-> ( ( y + 1 ) + 1 ) = ( 1 + ( y + 1 ) ) ) ) |
10 |
|
oveq1 |
|- ( x = A -> ( x + 1 ) = ( A + 1 ) ) |
11 |
|
oveq2 |
|- ( x = A -> ( 1 + x ) = ( 1 + A ) ) |
12 |
10 11
|
eqeq12d |
|- ( x = A -> ( ( x + 1 ) = ( 1 + x ) <-> ( A + 1 ) = ( 1 + A ) ) ) |
13 |
|
eqid |
|- ( 1 + 1 ) = ( 1 + 1 ) |
14 |
|
oveq1 |
|- ( ( y + 1 ) = ( 1 + y ) -> ( ( y + 1 ) + 1 ) = ( ( 1 + y ) + 1 ) ) |
15 |
|
1cnd |
|- ( y e. NN -> 1 e. CC ) |
16 |
|
nncn |
|- ( y e. NN -> y e. CC ) |
17 |
15 16 15
|
addassd |
|- ( y e. NN -> ( ( 1 + y ) + 1 ) = ( 1 + ( y + 1 ) ) ) |
18 |
14 17
|
sylan9eqr |
|- ( ( y e. NN /\ ( y + 1 ) = ( 1 + y ) ) -> ( ( y + 1 ) + 1 ) = ( 1 + ( y + 1 ) ) ) |
19 |
18
|
ex |
|- ( y e. NN -> ( ( y + 1 ) = ( 1 + y ) -> ( ( y + 1 ) + 1 ) = ( 1 + ( y + 1 ) ) ) ) |
20 |
3 6 9 12 13 19
|
nnind |
|- ( A e. NN -> ( A + 1 ) = ( 1 + A ) ) |