Step |
Hyp |
Ref |
Expression |
1 |
|
iftrue |
|- ( ph -> if ( ph , A , B ) = A ) |
2 |
|
iftrue |
|- ( ph -> if ( ph , C , D ) = C ) |
3 |
1 2
|
oveq12d |
|- ( ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = ( A F C ) ) |
4 |
|
iftrue |
|- ( ph -> if ( ph , ( A F C ) , ( B F D ) ) = ( A F C ) ) |
5 |
3 4
|
eqtr4d |
|- ( ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = if ( ph , ( A F C ) , ( B F D ) ) ) |
6 |
|
iffalse |
|- ( -. ph -> if ( ph , A , B ) = B ) |
7 |
|
iffalse |
|- ( -. ph -> if ( ph , C , D ) = D ) |
8 |
6 7
|
oveq12d |
|- ( -. ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = ( B F D ) ) |
9 |
|
iffalse |
|- ( -. ph -> if ( ph , ( A F C ) , ( B F D ) ) = ( B F D ) ) |
10 |
8 9
|
eqtr4d |
|- ( -. ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = if ( ph , ( A F C ) , ( B F D ) ) ) |
11 |
5 10
|
pm2.61i |
|- ( if ( ph , A , B ) F if ( ph , C , D ) ) = if ( ph , ( A F C ) , ( B F D ) ) |