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Theorem s5eqd 11531
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 11530 . . 3  |-  ( ph  ->  <" A B C D ">  =  <" N O P Q "> )
6 s5eqd.5 . . . 4  |-  ( ph  ->  E  =  R )
76s1eqd 11456 . . 3  |-  ( ph  ->  <" E ">  =  <" R "> )
85, 7oveq12d 5892 . 2  |-  ( ph  ->  ( <" A B C D "> concat  <" E "> )  =  ( <" N O P Q "> concat  <" R "> ) )
9 df-s5 11517 . 2  |-  <" A B C D E ">  =  ( <" A B C D "> concat  <" E "> )
10 df-s5 11517 . 2  |-  <" N O P Q R ">  =  ( <" N O P Q "> concat  <" R "> )
118, 9, 103eqtr4g 2353 1  |-  ( ph  ->  <" A B C D E ">  =  <" N O P Q R "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632  (class class class)co 5874   concat cconcat 11420   <"cs1 11421   <"cs4 11509   <"cs5 11510
This theorem is referenced by:  s6eqd  11532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-s1 11427  df-s2 11514  df-s3 11515  df-s4 11516  df-s5 11517
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