![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > deceq2 | GIF version |
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
deceq2 | ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5876 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵)) | |
2 | df-dec 9361 | . 2 ⊢ ;𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴) | |
3 | df-dec 9361 | . 2 ⊢ ;𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2235 | 1 ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 (class class class)co 5868 1c1 7790 + caddc 7792 · cmul 7794 9c9 8953 ;cdc 9360 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-iota 5173 df-fv 5219 df-ov 5871 df-dec 9361 |
This theorem is referenced by: deceq2i 9367 |
Copyright terms: Public domain | W3C validator |