MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s5eqd Unicode version

Theorem s5eqd 11515
Description: Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1  |-  ( ph  ->  A  =  N )
s2eqd.2  |-  ( ph  ->  B  =  O )
s3eqd.3  |-  ( ph  ->  C  =  P )
s4eqd.4  |-  ( ph  ->  D  =  Q )
s5eqd.5  |-  ( ph  ->  E  =  R )
Assertion
Ref Expression
s5eqd  |-  ( ph  ->  <" A B C D E ">  =  <" N O P Q R "> )

Proof of Theorem s5eqd
StepHypRef Expression
1 s2eqd.1 . . . 4  |-  ( ph  ->  A  =  N )
2 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
3 s3eqd.3 . . . 4  |-  ( ph  ->  C  =  P )
4 s4eqd.4 . . . 4  |-  ( ph  ->  D  =  Q )
51, 2, 3, 4s4eqd 11514 . . 3  |-  ( ph  ->  <" A B C D ">  =  <" N O P Q "> )
6 s5eqd.5 . . . 4  |-  ( ph  ->  E  =  R )
76s1eqd 11440 . . 3  |-  ( ph  ->  <" E ">  =  <" R "> )
85, 7oveq12d 5876 . 2  |-  ( ph  ->  ( <" A B C D "> concat  <" E "> )  =  ( <" N O P Q "> concat  <" R "> ) )
9 df-s5 11501 . 2  |-  <" A B C D E ">  =  ( <" A B C D "> concat  <" E "> )
10 df-s5 11501 . 2  |-  <" N O P Q R ">  =  ( <" N O P Q "> concat  <" R "> )
118, 9, 103eqtr4g 2340 1  |-  ( ph  ->  <" A B C D E ">  =  <" N O P Q R "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623  (class class class)co 5858   concat cconcat 11404   <"cs1 11405   <"cs4 11493   <"cs5 11494
This theorem is referenced by:  s6eqd  11516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-s1 11411  df-s2 11498  df-s3 11499  df-s4 11500  df-s5 11501
  Copyright terms: Public domain W3C validator