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Mirrors > Home > ILE Home > Th. List > opabbid | GIF version |
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
opabbid.1 | ⊢ Ⅎ𝑥𝜑 |
opabbid.2 | ⊢ Ⅎ𝑦𝜑 |
opabbid.3 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opabbid | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabbid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | opabbid.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | opabbid.3 | . . . . . 6 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 3 | anbi2d 437 | . . . . 5 ⊢ (𝜑 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
5 | 2, 4 | exbid 1507 | . . . 4 ⊢ (𝜑 → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
6 | 1, 5 | exbid 1507 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
7 | 6 | abbidv 2155 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒)}) |
8 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
9 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜒} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒)} | |
10 | 7, 8, 9 | 3eqtr4g 2097 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∃wex 1381 {cab 2026 〈cop 3378 {copab 3817 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-opab 3819 |
This theorem is referenced by: opabbidv 3823 mpteq12f 3837 fnoprabg 5602 |
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