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Mirrors > Home > ILE Home > Th. List > oprabbid | GIF version |
Description: Equivalent wff's yield equal operation class abstractions (deduction form). (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 461 | . . . . . 6 ⊢ (𝜑 → ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
6 | 3, 5 | exbid 1609 | . . . . 5 ⊢ (𝜑 → (∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
7 | 2, 6 | exbid 1609 | . . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
8 | 1, 7 | exbid 1609 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
9 | 8 | abbidv 2288 | . 2 ⊢ (𝜑 → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)}) |
10 | df-oprab 5857 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
11 | df-oprab 5857 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)} | |
12 | 9, 10, 11 | 3eqtr4g 2228 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 Ⅎwnf 1453 ∃wex 1485 {cab 2156 〈cop 3586 {coprab 5854 |
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-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-oprab 5857 |
This theorem is referenced by: oprabbidv 5907 mpoeq123 5912 |
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