Theorem deceq2 12745

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 7422	. 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵))
2		df-dec 12740	. 2 ⊢ ;𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴)
3		df-dec 12740	. 2 ⊢ ;𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵)
4	1, 2, 3	3eqtr4g 2820	1 ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  (class class class)co 7414  1c1 11128   + caddc 11130   · cmul 11132  9c9 12329  ;cdc 12739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-dec 12740

This theorem is used by:  deceq2i  12747