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Theorem s4eqd 11833
Description: Equality theorem for a length 4 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 )
Assertion
Ref Expression
s4eqd  |-  ( ph  ->  <" A B C D ">  =  <" N O P Q "> )

Proof of Theorem s4eqd
StepHypRef Expression
1 s2eqd.1 . . . 4  |-  ( ph  ->  A  =  N )
2 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
3 s3eqd.3 . . . 4  |-  ( ph  ->  C  =  P )
41, 2, 3s3eqd 11832 . . 3  |-  ( ph  ->  <" A B C ">  =  <" N O P "> )
5 s4eqd.4 . . . 4  |-  ( ph  ->  D  =  Q )
65s1eqd 11759 . . 3  |-  ( ph  ->  <" D ">  =  <" Q "> )
74, 6oveq12d 6102 . 2  |-  ( ph  ->  ( <" A B C "> concat  <" D "> )  =  (
<" N O P "> concat  <" Q "> ) )
8 df-s4 11819 . 2  |-  <" A B C D ">  =  ( <" A B C "> concat  <" D "> )
9 df-s4 11819 . 2  |-  <" N O P Q ">  =  ( <" N O P "> concat  <" Q "> )
107, 8, 93eqtr4g 2495 1  |-  ( ph  ->  <" A B C D ">  =  <" N O P Q "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653  (class class class)co 6084   concat cconcat 11723   <"cs1 11724   <"cs3 11811   <"cs4 11812
This theorem is referenced by:  s5eqd  11834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-s1 11730  df-s2 11817  df-s3 11818  df-s4 11819
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