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Theorem deceq2 12687
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
Assertion
Ref Expression
deceq2 (๐ด = ๐ต โ†’ ๐ถ๐ด = ๐ถ๐ต)

Proof of Theorem deceq2
StepHypRef Expression
1 oveq2 7419 . 2 (๐ด = ๐ต โ†’ (((9 + 1) ยท ๐ถ) + ๐ด) = (((9 + 1) ยท ๐ถ) + ๐ต))
2 df-dec 12682 . 2 ๐ถ๐ด = (((9 + 1) ยท ๐ถ) + ๐ด)
3 df-dec 12682 . 2 ๐ถ๐ต = (((9 + 1) ยท ๐ถ) + ๐ต)
41, 2, 33eqtr4g 2795 1 (๐ด = ๐ต โ†’ ๐ถ๐ด = ๐ถ๐ต)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1539  (class class class)co 7411  1c1 11113   + caddc 11115   ยท cmul 11117  9c9 12278  cdc 12681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-dec 12682
This theorem is referenced by:  deceq2i  12689
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