Theorem s6eqd

Description: Equality theorem for a length 6 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 )
Assertion	s6eqd	|- ( ph -> <" A B C D E F "> = <" N O P Q R S "> )

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	1 2 3 4 5	s5eqd	|- ( ph -> <" A B C D E "> = <" N O P Q R "> )
8	6	s1eqd	|- ( ph -> <" F "> = <" S "> )
9	7 8	oveq12d	|- ( ph -> ( <" A B C D E "> ++ <" F "> ) = ( <" N O P Q R "> ++ <" S "> ) )
10		df-s6	|- <" A B C D E F "> = ( <" A B C D E "> ++ <" F "> )
11		df-s6	|- <" N O P Q R S "> = ( <" N O P Q R "> ++ <" S "> )
12	9 10 11	3eqtr4g	|- ( ph -> <" A B C D E F "> = <" N O P Q R S "> )