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Mirrors > Home > ILE Home > Th. List > oveq | GIF version |
Description: Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Ref | Expression |
---|---|
oveq | ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5197 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹‘〈𝐴, 𝐵〉) = (𝐺‘〈𝐴, 𝐵〉)) | |
2 | df-ov 5535 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | df-ov 5535 | . 2 ⊢ (𝐴𝐺𝐵) = (𝐺‘〈𝐴, 𝐵〉) | |
4 | 1, 2, 3 | 3eqtr4g 2138 | 1 ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 〈cop 3401 ‘cfv 4922 (class class class)co 5532 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: oveqi 5545 oveqd 5549 ovmpt2df 5652 ovmpt2dv2 5654 |
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