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Mirrors > Home > ILE Home > Th. List > oveq123i | GIF version |
Description: Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
Ref | Expression |
---|---|
oveq123i.1 | ⊢ 𝐴 = 𝐶 |
oveq123i.2 | ⊢ 𝐵 = 𝐷 |
oveq123i.3 | ⊢ 𝐹 = 𝐺 |
Ref | Expression |
---|---|
oveq123i | ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq123i.1 | . . 3 ⊢ 𝐴 = 𝐶 | |
2 | oveq123i.2 | . . 3 ⊢ 𝐵 = 𝐷 | |
3 | 1, 2 | oveq12i 5664 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐶𝐹𝐷) |
4 | oveq123i.3 | . . 3 ⊢ 𝐹 = 𝐺 | |
5 | 4 | oveqi 5665 | . 2 ⊢ (𝐶𝐹𝐷) = (𝐶𝐺𝐷) |
6 | 3, 5 | eqtri 2108 | 1 ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 (class class class)co 5652 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-iota 4980 df-fv 5023 df-ov 5655 |
This theorem is referenced by: (None) |
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