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Theorem s2eqd 11826
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 11754 . . 3  |-  ( ph  ->  <" A ">  =  <" N "> )
3 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
43s1eqd 11754 . . 3  |-  ( ph  ->  <" B ">  =  <" O "> )
52, 4oveq12d 6099 . 2  |-  ( ph  ->  ( <" A "> concat  <" B "> )  =  ( <" N "> concat  <" O "> ) )
6 df-s2 11812 . 2  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
7 df-s2 11812 . 2  |-  <" N O ">  =  (
<" N "> concat  <" O "> )
85, 6, 73eqtr4g 2493 1  |-  ( ph  ->  <" A B ">  =  <" N O "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652  (class class class)co 6081   concat cconcat 11718   <"cs1 11719   <"cs2 11805
This theorem is referenced by:  s3eqd  11827  efgi  15351  efgi0  15352  efgi1  15353  efgtf  15354  efgtval  15355  efgval2  15356  frgpuplem  15404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-s1 11725  df-s2 11812
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