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Theorem s4eqd 11783
Description: Equality theorem for a length 4 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 )
Assertion
Ref Expression
s4eqd  |-  ( ph  ->  <" A B C D ">  =  <" N O P Q "> )

Proof of Theorem s4eqd
StepHypRef Expression
1 s2eqd.1 . . . 4  |-  ( ph  ->  A  =  N )
2 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
3 s3eqd.3 . . . 4  |-  ( ph  ->  C  =  P )
41, 2, 3s3eqd 11782 . . 3  |-  ( ph  ->  <" A B C ">  =  <" N O P "> )
5 s4eqd.4 . . . 4  |-  ( ph  ->  D  =  Q )
65s1eqd 11709 . . 3  |-  ( ph  ->  <" D ">  =  <" Q "> )
74, 6oveq12d 6058 . 2  |-  ( ph  ->  ( <" A B C "> concat  <" D "> )  =  (
<" N O P "> concat  <" Q "> ) )
8 df-s4 11769 . 2  |-  <" A B C D ">  =  ( <" A B C "> concat  <" D "> )
9 df-s4 11769 . 2  |-  <" N O P Q ">  =  ( <" N O P "> concat  <" Q "> )
107, 8, 93eqtr4g 2461 1  |-  ( ph  ->  <" A B C D ">  =  <" N O P Q "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649  (class class class)co 6040   concat cconcat 11673   <"cs1 11674   <"cs3 11761   <"cs4 11762
This theorem is referenced by:  s5eqd  11784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-s1 11680  df-s2 11767  df-s3 11768  df-s4 11769
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