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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 5791 | . 2 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐴𝐺𝐶)) |
3 | oveq123d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | oveq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 3, 4 | oveq12d 5792 | . 2 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐵𝐺𝐷)) |
6 | 2, 5 | eqtrd 2172 | 1 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 (class class class)co 5774 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: csbov123g 5809 |
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