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Mirrors > Home > ILE Home > Th. List > eloprabg | GIF version |
Description: The law of concretion for operation class abstraction. Compare elopab 4180. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
eloprabg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
eloprabg.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
eloprabg.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
eloprabg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloprabg.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | eloprabg.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | eloprabg.3 | . . 3 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
4 | 1, 2, 3 | syl3an9b 1288 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜃)) |
5 | 4 | eloprabga 5858 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 〈cop 3530 {coprab 5775 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
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-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-oprab 5778 |
This theorem is referenced by: ov 5890 ovg 5909 |
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