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Theorem s3eqd 11819
Description: Equality theorem for a length 3 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 )
Assertion
Ref Expression
s3eqd  |-  ( ph  ->  <" A B C ">  =  <" N O P "> )

Proof of Theorem s3eqd
StepHypRef Expression
1 s2eqd.1 . . . 4  |-  ( ph  ->  A  =  N )
2 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
31, 2s2eqd 11818 . . 3  |-  ( ph  ->  <" A B ">  =  <" N O "> )
4 s3eqd.3 . . . 4  |-  ( ph  ->  C  =  P )
54s1eqd 11746 . . 3  |-  ( ph  ->  <" C ">  =  <" P "> )
63, 5oveq12d 6091 . 2  |-  ( ph  ->  ( <" A B "> concat  <" C "> )  =  (
<" N O "> concat 
<" P "> ) )
7 df-s3 11805 . 2  |-  <" A B C ">  =  ( <" A B "> concat  <" C "> )
8 df-s3 11805 . 2  |-  <" N O P ">  =  ( <" N O "> concat  <" P "> )
96, 7, 83eqtr4g 2492 1  |-  ( ph  ->  <" A B C ">  =  <" N O P "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652  (class class class)co 6073   concat cconcat 11710   <"cs1 11711   <"cs2 11797   <"cs3 11798
This theorem is referenced by:  s4eqd  11820
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-s1 11717  df-s2 11804  df-s3 11805
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