Theorem deceq2 9419

Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)

Assertion

Ref	Expression
deceq2	|- ( A = B -> ; C A = ; C B )

Proof of Theorem deceq2

Step	Hyp	Ref	Expression
1		oveq2 5904	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. C ) + A ) = ( ( ( 9 + 1 ) x. C ) + B ) )
2		df-dec 9415	. 2 |- ; C A = ( ( ( 9 + 1 ) x. C ) + A )
3		df-dec 9415	. 2 |- ; C B = ( ( ( 9 + 1 ) x. C ) + B )
4	1, 2, 3	3eqtr4g 2247	1 |- ( A = B -> ; C A = ; C B )

Syntax hints:   -> wi 4   = wceq 1364  (class class class)co 5896   1c1 7842   + caddc 7844   x. cmul 7846   9c9 9007  ;cdc 9414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899  df-dec 9415

This theorem is referenced by:  deceq2i  9421