Theorem dp2eq1 33052

Description: Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

Ref	Expression
dp2eq1	⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)

Proof of Theorem dp2eq1

Step	Hyp	Ref	Expression
1		oveq1 7405	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 + (𝐶 / ;10)) = (𝐵 + (𝐶 / ;10)))
2		df-dp2 33051	. 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10))
3		df-dp2 33051	. 2 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10))
4	1, 2, 3	3eqtr4g 2824	1 ⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1562  (class class class)co 7398  0cc0 11075  1c1 11076   + caddc 11078   / cdiv 11846  ;cdc 12690  _cdp2 33050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-dp2 33051

This theorem is referenced by:  dp2eq1i  33054