Theorem deceq2 9583

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 → ;𝐶𝐴 = ;𝐶𝐵)

Proof of Theorem deceq2

Step	Hyp	Ref	Expression
1		oveq2 6009	. 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵))
2		df-dec 9579	. 2 ⊢ ;𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴)
3		df-dec 9579	. 2 ⊢ ;𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵)
4	1, 2, 3	3eqtr4g 2287	1 ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵)

Syntax hints:   → wi 4   = wceq 1395  (class class class)co 6001  1c1 8000   + caddc 8002   · cmul 8004  9c9 9168  ;cdc 9578

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004  df-dec 9579

This theorem is referenced by:  deceq2i  9585