Theorem deceq2 9456

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 5927	. 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵))
2		df-dec 9452	. 2 ⊢ ;𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴)
3		df-dec 9452	. 2 ⊢ ;𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵)
4	1, 2, 3	3eqtr4g 2251	1 ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵)

Syntax hints:   → wi 4   = wceq 1364  (class class class)co 5919  1c1 7875   + caddc 7877   · cmul 7879  9c9 9042  ;cdc 9451

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 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922  df-dec 9452

This theorem is referenced by:  deceq2i  9458