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Theorem s8eqd 12966
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 12965 . . 3  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )
9 s8eqd.6 . . . 4  |-  ( ph  ->  H  =  U )
109s1eqd 12744 . . 3  |-  ( ph  ->  <" H ">  =  <" U "> )
118, 10oveq12d 6323 . 2  |-  ( ph  ->  ( <" A B C D E F G "> ++  <" H "> )  =  (
<" N O P Q R S T "> ++  <" U "> ) )
12 df-s8 12952 . 2  |-  <" A B C D E F G H ">  =  ( <" A B C D E F G "> ++  <" H "> )
13 df-s8 12952 . 2  |-  <" N O P Q R S T U ">  =  ( <" N O P Q R S T "> ++  <" U "> )
1411, 12, 133eqtr4g 2488 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 1437  (class class class)co 6305   ++ cconcat 12662   <"cs1 12663   <"cs7 12944   <"cs8 12945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-iota 5565  df-fv 5609  df-ov 6308  df-s1 12671  df-s2 12946  df-s3 12947  df-s4 12948  df-s5 12949  df-s6 12950  df-s7 12951  df-s8 12952
This theorem is referenced by: (None)
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