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Theorem s2eqd 12485
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 12288 . . 3  |-  ( ph  ->  <" A ">  =  <" N "> )
3 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
43s1eqd 12288 . . 3  |-  ( ph  ->  <" B ">  =  <" O "> )
52, 4oveq12d 6108 . 2  |-  ( ph  ->  ( <" A "> concat  <" B "> )  =  ( <" N "> concat  <" O "> ) )
6 df-s2 12471 . 2  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
7 df-s2 12471 . 2  |-  <" N O ">  =  (
<" N "> concat  <" O "> )
85, 6, 73eqtr4g 2498 1  |-  ( ph  ->  <" A B ">  =  <" N O "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364  (class class class)co 6090   concat cconcat 12219   <"cs1 12220   <"cs2 12464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-ov 6093  df-s1 12228  df-s2 12471
This theorem is referenced by:  s3eqd  12486  swrd2lsw  12548  efgi  16209  efgi0  16210  efgi1  16211  efgtf  16212  efgtval  16213  efgval2  16214  frgpuplem  16262
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