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Theorem s7eqd 12959
Description: Equality theorem for a length 7 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 )
Assertion
Ref Expression
s7eqd  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )

Proof of Theorem s7eqd
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 )
71, 2, 3, 4, 5, 6s6eqd 12958 . . 3  |-  ( ph  ->  <" A B C D E F ">  =  <" N O P Q R S "> )
8 s7eqd.6 . . . 4  |-  ( ph  ->  G  =  T )
98s1eqd 12738 . . 3  |-  ( ph  ->  <" G ">  =  <" T "> )
107, 9oveq12d 6321 . 2  |-  ( ph  ->  ( <" A B C D E F "> ++  <" G "> )  =  (
<" N O P Q R S "> ++  <" T "> ) )
11 df-s7 12945 . 2  |-  <" A B C D E F G ">  =  ( <" A B C D E F "> ++  <" G "> )
12 df-s7 12945 . 2  |-  <" N O P Q R S T ">  =  ( <" N O P Q R S "> ++  <" T "> )
1310, 11, 123eqtr4g 2489 1  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438  (class class class)co 6303   ++ cconcat 12656   <"cs1 12657   <"cs6 12937   <"cs7 12938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-iota 5563  df-fv 5607  df-ov 6306  df-s1 12665  df-s2 12940  df-s3 12941  df-s4 12942  df-s5 12943  df-s6 12944  df-s7 12945
This theorem is referenced by:  s8eqd  12960
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