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Theorem s3eqd 11513
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 11512 . . 3  |-  ( ph  ->  <" A B ">  =  <" N O "> )
4 s3eqd.3 . . . 4  |-  ( ph  ->  C  =  P )
54s1eqd 11440 . . 3  |-  ( ph  ->  <" C ">  =  <" P "> )
63, 5oveq12d 5876 . 2  |-  ( ph  ->  ( <" A B "> concat  <" C "> )  =  (
<" N O "> concat 
<" P "> ) )
7 df-s3 11499 . 2  |-  <" A B C ">  =  ( <" A B "> concat  <" C "> )
8 df-s3 11499 . 2  |-  <" N O P ">  =  ( <" N O "> concat  <" P "> )
96, 7, 83eqtr4g 2340 1  |-  ( ph  ->  <" A B C ">  =  <" N O P "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623  (class class class)co 5858   concat cconcat 11404   <"cs1 11405   <"cs2 11491   <"cs3 11492
This theorem is referenced by:  s4eqd  11514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-s1 11411  df-s2 11498  df-s3 11499
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