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Theorem s2eqd 12806
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 12592 . . 3  |-  ( ph  ->  <" A ">  =  <" N "> )
3 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
43s1eqd 12592 . . 3  |-  ( ph  ->  <" B ">  =  <" O "> )
52, 4oveq12d 6299 . 2  |-  ( ph  ->  ( <" A "> concat  <" B "> )  =  ( <" N "> concat  <" O "> ) )
6 df-s2 12792 . 2  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
7 df-s2 12792 . 2  |-  <" N O ">  =  (
<" N "> concat  <" O "> )
85, 6, 73eqtr4g 2509 1  |-  ( ph  ->  <" A B ">  =  <" N O "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383  (class class class)co 6281   concat cconcat 12515   <"cs1 12516   <"cs2 12785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-s1 12524  df-s2 12792
This theorem is referenced by:  s3eqd  12807  swrd2lsw  12869  efgi  16611  efgi0  16612  efgi1  16613  efgtf  16614  efgtval  16615  efgval2  16616  frgpuplem  16664
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