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Mirrors > Home > ILE Home > Th. List > oveqd | GIF version |
Description: Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
Ref | Expression |
---|---|
oveq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
oveqd | ⊢ (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | oveq 5832 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶𝐴𝐷) = (𝐶𝐵𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 (class class class)co 5826 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-uni 3775 df-br 3968 df-iota 5137 df-fv 5180 df-ov 5829 |
This theorem is referenced by: oveq123d 5847 csbov12g 5862 ovmpodxf 5948 oprssov 5964 ofeq 6036 fnmpoovd 6164 seqeq2 10357 blfvalps 12855 |
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