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Theorem s5eqd 11829
Description: Equality theorem for a length 5 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 )
Assertion
Ref Expression
s5eqd  |-  ( ph  ->  <" A B C D E ">  =  <" N O P Q R "> )

Proof of Theorem s5eqd
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 )
51, 2, 3, 4s4eqd 11828 . . 3  |-  ( ph  ->  <" A B C D ">  =  <" N O P Q "> )
6 s5eqd.5 . . . 4  |-  ( ph  ->  E  =  R )
76s1eqd 11754 . . 3  |-  ( ph  ->  <" E ">  =  <" R "> )
85, 7oveq12d 6099 . 2  |-  ( ph  ->  ( <" A B C D "> concat  <" E "> )  =  ( <" N O P Q "> concat  <" R "> ) )
9 df-s5 11815 . 2  |-  <" A B C D E ">  =  ( <" A B C D "> concat  <" E "> )
10 df-s5 11815 . 2  |-  <" N O P Q R ">  =  ( <" N O P Q "> concat  <" R "> )
118, 9, 103eqtr4g 2493 1  |-  ( ph  ->  <" A B C D E ">  =  <" N O P Q R "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652  (class class class)co 6081   concat cconcat 11718   <"cs1 11719   <"cs4 11807   <"cs5 11808
This theorem is referenced by:  s6eqd  11830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-s1 11725  df-s2 11812  df-s3 11813  df-s4 11814  df-s5 11815
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