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Mirrors > Home > ILE Home > Th. List > oveq123d | GIF version |
Description: Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
Ref | Expression |
---|---|
oveq123d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
oveq123d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
oveq123d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
oveq123d | ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq123d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | 1 | oveqd 5549 | . 2 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐴𝐺𝐶)) |
3 | oveq123d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | oveq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 3, 4 | oveq12d 5550 | . 2 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐵𝐺𝐷)) |
6 | 2, 5 | eqtrd 2113 | 1 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 (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-3an 921 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-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: csbov123g 5563 |
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