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Mirrors > Home > ILE Home > Th. List > eloprabg | GIF version |
Description: The law of concretion for operation class abstraction. Compare elopab 4255. (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 1310 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜃)) |
5 | 4 | eloprabga 5956 | 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 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 〈cop 3594 {coprab 5870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-oprab 5873 |
This theorem is referenced by: ov 5988 ovg 6007 |
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