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Theorem s6eqd 11759
Description: Equality theorem for a length 6 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 )
Assertion
Ref Expression
s6eqd  |-  ( ph  ->  <" A B C D E F ">  =  <" N O P Q R S "> )

Proof of Theorem s6eqd
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 )
61, 2, 3, 4, 5s5eqd 11758 . . 3  |-  ( ph  ->  <" A B C D E ">  =  <" N O P Q R "> )
7 s6eqd.6 . . . 4  |-  ( ph  ->  F  =  S )
87s1eqd 11683 . . 3  |-  ( ph  ->  <" F ">  =  <" S "> )
96, 8oveq12d 6040 . 2  |-  ( ph  ->  ( <" A B C D E "> concat 
<" F "> )  =  ( <" N O P Q R "> concat  <" S "> ) )
10 df-s6 11745 . 2  |-  <" A B C D E F ">  =  (
<" A B C D E "> concat  <" F "> )
11 df-s6 11745 . 2  |-  <" N O P Q R S ">  =  (
<" N O P Q R "> concat  <" S "> )
129, 10, 113eqtr4g 2446 1  |-  ( ph  ->  <" A B C D E F ">  =  <" N O P Q R S "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649  (class class class)co 6022   concat cconcat 11647   <"cs1 11648   <"cs5 11737   <"cs6 11738
This theorem is referenced by:  s7eqd  11760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-ov 6025  df-s1 11654  df-s2 11741  df-s3 11742  df-s4 11743  df-s5 11744  df-s6 11745
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