Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > deceq1 | GIF version |
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
deceq1 | ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5844 | . . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵)) | |
2 | 1 | oveq1d 5851 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶)) |
3 | df-dec 9314 | . 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶) | |
4 | df-dec 9314 | . 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶) | |
5 | 2, 3, 4 | 3eqtr4g 2222 | 1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 (class class class)co 5836 1c1 7745 + caddc 7747 · cmul 7749 9c9 8906 ;cdc 9313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 df-dec 9314 |
This theorem is referenced by: deceq1i 9319 |
Copyright terms: Public domain | W3C validator |