Theorem deceq2 9362

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 5873	. 2 |- ( A = B -> ( ( ( 9 + 1 ) x. C ) + A ) = ( ( ( 9 + 1 ) x. C ) + B ) )
2		df-dec 9358	. 2 |- ; C A = ( ( ( 9 + 1 ) x. C ) + A )
3		df-dec 9358	. 2 |- ; C B = ( ( ( 9 + 1 ) x. C ) + B )
4	1, 2, 3	3eqtr4g 2233	1 |- ( A = B -> ; C A = ; C B )

Syntax hints:   -> wi 4   = wceq 1353  (class class class)co 5865   1c1 7787   + caddc 7789   x. cmul 7791   9c9 8950  ;cdc 9357

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868  df-dec 9358

This theorem is referenced by:  deceq2i  9364