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Theorem s2eqd 12489
Description: Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1  |-  ( ph  ->  A  =  N )
s2eqd.2  |-  ( ph  ->  B  =  O )
Assertion
Ref Expression
s2eqd  |-  ( ph  ->  <" A B ">  =  <" N O "> )

Proof of Theorem s2eqd
StepHypRef Expression
1 s2eqd.1 . . . 4  |-  ( ph  ->  A  =  N )
21s1eqd 12292 . . 3  |-  ( ph  ->  <" A ">  =  <" N "> )
3 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
43s1eqd 12292 . . 3  |-  ( ph  ->  <" B ">  =  <" O "> )
52, 4oveq12d 6109 . 2  |-  ( ph  ->  ( <" A "> concat  <" B "> )  =  ( <" N "> concat  <" O "> ) )
6 df-s2 12475 . 2  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
7 df-s2 12475 . 2  |-  <" N O ">  =  (
<" N "> concat  <" O "> )
85, 6, 73eqtr4g 2500 1  |-  ( ph  ->  <" A B ">  =  <" N O "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369  (class class class)co 6091   concat cconcat 12223   <"cs1 12224   <"cs2 12468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-s1 12232  df-s2 12475
This theorem is referenced by:  s3eqd  12490  swrd2lsw  12552  efgi  16216  efgi0  16217  efgi1  16218  efgtf  16219  efgtval  16220  efgval2  16221  frgpuplem  16269
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