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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 629 | . . . . . 6 ⊢ (𝜑 → ((𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ (𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
3 | 2 | exbidv 1924 | . . . . 5 ⊢ (𝜑 → (∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
4 | 3 | exbidv 1924 | . . . 4 ⊢ (𝜑 → (∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
5 | 4 | exbidv 1924 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒))) |
6 | 5 | abbidv 2800 | . 2 ⊢ (𝜑 → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)}) |
7 | df-oprab 7397 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜓)} | |
8 | df-oprab 7397 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜒)} | |
9 | 6, 7, 8 | 3eqtr4g 2796 | 1 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 {cab 2708 〈cop 4628 {coprab 7394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-oprab 7397 |
This theorem is referenced by: oprabbii 7460 mpoeq123dva 7467 mpoeq3dva 7470 resoprab2 7511 erovlem 8790 joinfval 18308 meetfval 18322 odujoin 18343 odumeet 18345 mppsval 34394 csbmpo123 36016 unceq 36269 uncf 36271 unccur 36275 |
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