Theorem deceq2 9462

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 5930	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. C ) + A ) = ( ( ( 9 + 1 ) x. C ) + B ) )
2		df-dec 9458	. 2 |- ; C A = ( ( ( 9 + 1 ) x. C ) + A )
3		df-dec 9458	. 2 |- ; C B = ( ( ( 9 + 1 ) x. C ) + B )
4	1, 2, 3	3eqtr4g 2254	1 |- ( A = B -> ; C A = ; C B )

Syntax hints:   -> wi 4   = wceq 1364  (class class class)co 5922   1c1 7880   + caddc 7882   x. cmul 7884   9c9 9048  ;cdc 9457

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-dec 9458

This theorem is referenced by:  deceq2i  9464