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Mirrors > Home > ILE Home > Th. List > oveqan12rd | GIF version |
Description: Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
Ref | Expression |
---|---|
oveq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
opreqan12i.2 | ⊢ (𝜓 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
oveqan12rd | ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | opreqan12i.2 | . . 3 ⊢ (𝜓 → 𝐶 = 𝐷) | |
3 | 1, 2 | oveqan12d 5801 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
4 | 3 | ancoms 266 | 1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = 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: mulresr 7670 recdivap 8502 divgcdcoprm0 11818 dvid 12870 |
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