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Theorem deceq1 10385
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
Assertion
Ref Expression
deceq1  |-  ( A  =  B  -> ; A C  = ; B C )

Proof of Theorem deceq1
StepHypRef Expression
1 oveq2 6089 . . 3  |-  ( A  =  B  ->  ( 10  x.  A )  =  ( 10  x.  B
) )
21oveq1d 6096 . 2  |-  ( A  =  B  ->  (
( 10  x.  A
)  +  C )  =  ( ( 10  x.  B )  +  C ) )
3 df-dec 10383 . 2  |- ; A C  =  ( ( 10  x.  A
)  +  C )
4 df-dec 10383 . 2  |- ; B C  =  ( ( 10  x.  B
)  +  C )
52, 3, 43eqtr4g 2493 1  |-  ( A  =  B  -> ; A C  = ; B C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652  (class class class)co 6081    + caddc 8993    x. cmul 8995   10c10 10057  ;cdc 10382
This theorem is referenced by:  deceq1i  10387
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-dec 10383
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