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| Mirrors > Home > MPE Home > Th. List > oprabbidv | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) |
| Ref | Expression |
|---|---|
| oprabbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| oprabbidv | ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oprabbidv.1 | . . . . . . 7 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | anbi2d 630 | . . . . . 6 ⊢ (𝜑 → ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
| 3 | 2 | exbidv 1921 | . . . . 5 ⊢ (𝜑 → (∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
| 4 | 3 | exbidv 1921 | . . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
| 5 | 4 | exbidv 1921 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
| 6 | 5 | abbidv 2808 | . 2 ⊢ (𝜑 → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)}) |
| 7 | df-oprab 7435 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
| 8 | df-oprab 7435 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)} | |
| 9 | 6, 7, 8 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 {cab 2714 〈cop 4632 {coprab 7432 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-oprab 7435 |
| This theorem is referenced by: oprabbii 7500 mpoeq123dva 7507 mpoeq3dva 7510 resoprab2 7552 erovlem 8853 joinfval 18418 meetfval 18432 odujoin 18453 odumeet 18455 mppsval 35577 csbmpo123 37332 unceq 37604 uncf 37606 unccur 37610 |
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