MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s2eqd Structured version   Visualization version   Unicode version

Theorem s2eqd 13018
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 12793 . . 3  |-  ( ph  ->  <" A ">  =  <" N "> )
3 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
43s1eqd 12793 . . 3  |-  ( ph  ->  <" B ">  =  <" O "> )
52, 4oveq12d 6326 . 2  |-  ( ph  ->  ( <" A "> ++  <" B "> )  =  ( <" N "> ++  <" O "> ) )
6 df-s2 13003 . 2  |-  <" A B ">  =  (
<" A "> ++  <" B "> )
7 df-s2 13003 . 2  |-  <" N O ">  =  (
<" N "> ++  <" O "> )
85, 6, 73eqtr4g 2530 1  |-  ( ph  ->  <" A B ">  =  <" N O "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452  (class class class)co 6308   ++ cconcat 12705   <"cs1 12706   <"cs2 12996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-ov 6311  df-s1 12714  df-s2 13003
This theorem is referenced by:  s3eqd  13019  wrdl2exs2  13097  swrd2lsw  13102  efgi  17447  efgi0  17448  efgi1  17449  efgtf  17450  efgtval  17451  efgval2  17452  frgpuplem  17500
  Copyright terms: Public domain W3C validator