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Mirrors > Home > ILE Home > Th. List > oprabbid | GIF version |
Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
oprabbid.1 | ⊢ Ⅎ𝑥𝜑 |
oprabbid.2 | ⊢ Ⅎ𝑦𝜑 |
oprabbid.3 | ⊢ Ⅎ𝑧𝜑 |
oprabbid.4 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
oprabbid | ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oprabbid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | oprabbid.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | oprabbid.3 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | oprabbid.4 | . . . . . . 7 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 4 | anbi2d 452 | . . . . . 6 ⊢ (𝜑 → ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
6 | 3, 5 | exbid 1548 | . . . . 5 ⊢ (𝜑 → (∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
7 | 2, 6 | exbid 1548 | . . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
8 | 1, 7 | exbid 1548 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
9 | 8 | abbidv 2200 | . 2 ⊢ (𝜑 → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)}) |
10 | df-oprab 5595 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
11 | df-oprab 5595 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)} | |
12 | 9, 10, 11 | 3eqtr4g 2140 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 Ⅎwnf 1390 ∃wex 1422 {cab 2069 〈cop 3425 {coprab 5592 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-11 1438 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-oprab 5595 |
This theorem is referenced by: oprabbidv 5638 mpt2eq123 5643 |
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