Theorem dp2eq1 29580

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 6657	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 + (𝐶 / ;10)) = (𝐵 + (𝐶 / ;10)))
2		df-dp2 29578	. 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10))
3		df-dp2 29578	. 2 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10))
4	1, 2, 3	3eqtr4g 2681	1 ⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1483  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939   / cdiv 10684  ;cdc 11493  _cdp2 29577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-dp2 29578

This theorem is referenced by:  dp2eq1i  29582