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Theorem s8eqd 12887
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
StepHypRef 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 )
81, 2, 3, 4, 5, 6, 7s7eqd 12886 . . 3  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )
9 s8eqd.6 . . . 4  |-  ( ph  ->  H  =  U )
109s1eqd 12665 . . 3  |-  ( ph  ->  <" H ">  =  <" U "> )
118, 10oveq12d 6295 . 2  |-  ( ph  ->  ( <" A B C D E F G "> ++  <" H "> )  =  (
<" N O P Q R S T "> ++  <" U "> ) )
12 df-s8 12873 . 2  |-  <" A B C D E F G H ">  =  ( <" A B C D E F G "> ++  <" H "> )
13 df-s8 12873 . 2  |-  <" N O P Q R S T U ">  =  ( <" N O P Q R S T "> ++  <" U "> )
1411, 12, 133eqtr4g 2468 1  |-  ( ph  ->  <" A B C D E F G H ">  =  <" N O P Q R S T U "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405  (class class class)co 6277   ++ cconcat 12583   <"cs1 12584   <"cs7 12865   <"cs8 12866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-s1 12592  df-s2 12867  df-s3 12868  df-s4 12869  df-s5 12870  df-s6 12871  df-s7 12872  df-s8 12873
This theorem is referenced by: (None)
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