Theorem s8eqd 11548

Description: Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
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 )
s8eqd.6	|- ( ph -> H = U )

Assertion

Ref	Expression
s8eqd	|- ( ph -> <" A B C D E F G H "> = <" N O P Q R S T U "> )

Proof of Theorem s8eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 |- ( ph -> A = N )
2		s2eqd.2	. . . 4 |- ( ph -> B = O )
3		s3eqd.3	. . . 4 |- ( ph -> C = P )
4		s4eqd.4	. . . 4 |- ( ph -> D = Q )
5		s5eqd.5	. . . 4 |- ( ph -> E = R )
6		s6eqd.6	. . . 4 |- ( ph -> F = S )
7		s7eqd.6	. . . 4 |- ( ph -> G = T )
8	1, 2, 3, 4, 5, 6, 7	s7eqd 11547	. . 3 |- ( ph -> <" A B C D E F G "> = <" N O P Q R S T "> )
9		s8eqd.6	. . . 4 |- ( ph -> H = U )
10	9	s1eqd 11388	. . 3 |- ( ph -> <" H "> = <" U "> )
11	8, 10	oveq12d 6103	. 2 |- ( ph -> ( <" A B C D E F G "> ++ <" H "> ) = ( <" N O P Q R S T "> ++ <" U "> ) )
12		df-s8 11534	. 2 |- <" A B C D E F G H "> = ( <" A B C D E F G "> ++ <" H "> )
13		df-s8 11534	. 2 |- <" N O P Q R S T U "> = ( <" N O P Q R S T "> ++ <" U "> )
14	11, 12, 13	3eqtr4g 2296	1 |- ( ph -> <" A B C D E F G H "> = <" N O P Q R S T U "> )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402  (class class class)co 6085   ++ cconcat 11358   <"cs1 11383   <"cs7 11526   <"cs8 11527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-s1 11384  df-s2 11528  df-s3 11529  df-s4 11530  df-s5 11531  df-s6 11532  df-s7 11533  df-s8 11534

This theorem is used by: (None)