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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 5799 | . 2 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐴𝐺𝐶)) |
3 | oveq123d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | oveq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 3, 4 | oveq12d 5800 | . 2 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐵𝐺𝐷)) |
6 | 2, 5 | eqtrd 2173 | 1 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 (class class class)co 5782 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: csbov123g 5817 |
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