Description: Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | s2eqd.1 | |- ( ph -> A = N ) |
|
s2eqd.2 | |- ( ph -> B = O ) |
||
s3eqd.3 | |- ( ph -> C = P ) |
||
Assertion | s3eqd | |- ( ph -> <" A B C "> = <" N O P "> ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s2eqd.1 | |- ( ph -> A = N ) |
|
2 | s2eqd.2 | |- ( ph -> B = O ) |
|
3 | s3eqd.3 | |- ( ph -> C = P ) |
|
4 | 1 2 | s2eqd | |- ( ph -> <" A B "> = <" N O "> ) |
5 | 3 | s1eqd | |- ( ph -> <" C "> = <" P "> ) |
6 | 4 5 | oveq12d | |- ( ph -> ( <" A B "> ++ <" C "> ) = ( <" N O "> ++ <" P "> ) ) |
7 | df-s3 | |- <" A B C "> = ( <" A B "> ++ <" C "> ) |
|
8 | df-s3 | |- <" N O P "> = ( <" N O "> ++ <" P "> ) |
|
9 | 6 7 8 | 3eqtr4g | |- ( ph -> <" A B C "> = <" N O P "> ) |