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

Theorem s2eqd 11528
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 11456 . . 3  |-  ( ph  ->  <" A ">  =  <" N "> )
3 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
43s1eqd 11456 . . 3  |-  ( ph  ->  <" B ">  =  <" O "> )
52, 4oveq12d 5892 . 2  |-  ( ph  ->  ( <" A "> concat  <" B "> )  =  ( <" N "> concat  <" O "> ) )
6 df-s2 11514 . 2  |-  <" A B ">  =  (
<" A "> concat  <" B "> )
7 df-s2 11514 . 2  |-  <" N O ">  =  (
<" N "> concat  <" O "> )
85, 6, 73eqtr4g 2353 1  |-  ( ph  ->  <" A B ">  =  <" N O "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632  (class class class)co 5874   concat cconcat 11420   <"cs1 11421   <"cs2 11507
This theorem is referenced by:  s3eqd  11529  efgi  15044  efgi0  15045  efgi1  15046  efgtf  15047  efgtval  15048  efgval2  15049  frgpuplem  15097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-s1 11427  df-s2 11514
  Copyright terms: Public domain W3C validator