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Theorem s8eqd 12783
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 12782 . . 3  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )
9 s8eqd.6 . . . 4  |-  ( ph  ->  H  =  U )
109s1eqd 12563 . . 3  |-  ( ph  ->  <" H ">  =  <" U "> )
118, 10oveq12d 6293 . 2  |-  ( ph  ->  ( <" A B C D E F G "> concat  <" H "> )  =  (
<" N O P Q R S T "> concat  <" U "> ) )
12 df-s8 12769 . 2  |-  <" A B C D E F G H ">  =  ( <" A B C D E F G "> concat  <" H "> )
13 df-s8 12769 . 2  |-  <" N O P Q R S T U ">  =  ( <" N O P Q R S T "> concat  <" U "> )
1411, 12, 133eqtr4g 2526 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 1374  (class class class)co 6275   concat cconcat 12489   <"cs1 12490   <"cs7 12761   <"cs8 12762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-ov 6278  df-s1 12498  df-s2 12763  df-s3 12764  df-s4 12765  df-s5 12766  df-s6 12767  df-s7 12768  df-s8 12769
This theorem is referenced by: (None)
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