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Metamath Proof Explorer

Theorem s7eqd

Description: Equality theorem for a length 7 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 )
s4eqd.4 
|- ( ph -> D = Q )
s5eqd.5 
|- ( ph -> E = R )
s6eqd.6 
|- ( ph -> F = S )
s7eqd.6 
|- ( ph -> G = T )
Assertion s7eqd 
|- ( ph -> <" A B C D E F G "> = <" N O P Q R S T "> )

Proof

Step Hyp Ref Expression
1 s2eqd.1 
 |-  ( ph -> A = N )
2 s2eqd.2 
 |-  ( ph -> B = O )
3 s3eqd.3 
 |-  ( ph -> C = P )
4 s4eqd.4 
 |-  ( ph -> D = Q )
5 s5eqd.5 
 |-  ( ph -> E = R )
6 s6eqd.6 
 |-  ( ph -> F = S )
7 s7eqd.6 
 |-  ( ph -> G = T )
8 1 2 3 4 5 6 s6eqd 
 |-  ( ph -> <" A B C D E F "> = <" N O P Q R S "> )
9 7 s1eqd 
 |-  ( ph -> <" G "> = <" T "> )
10 8 9 oveq12d 
 |-  ( ph -> ( <" A B C D E F "> ++ <" G "> ) = ( <" N O P Q R S "> ++ <" T "> ) )
11 df-s7 
 |-  <" A B C D E F G "> = ( <" A B C D E F "> ++ <" G "> )
12 df-s7 
 |-  <" N O P Q R S T "> = ( <" N O P Q R S "> ++ <" T "> )
13 10 11 12 3eqtr4g 
 |-  ( ph -> <" A B C D E F G "> = <" N O P Q R S T "> )