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| Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5529 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2376. (Revised by GG, 26-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| opabidw | ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opex 5468 | . 2 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 2 | copsexgw 5494 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | |
| 3 | 2 | bicomd 223 | . 2 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑)) | 
| 4 | df-opab 5205 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 5 | 1, 3, 4 | elab2 3681 | 1 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 〈cop 4631 {copab 5204 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 | 
| This theorem is referenced by: rexopabb 5532 ssopab2bw 5551 dmopab 5925 rnopab 5964 funopab 6600 opabiota 6990 fvopab5 7048 f1ompt 7130 ovid 7575 zfrep6 7980 enssdom 9018 omxpenlem 9114 infxpenlem 10054 canthwelem 10691 pospo 18391 2ndcdisj 23465 lgsquadlem1 27425 lgsquadlem2 27426 h2hlm 31000 opabdm 32624 opabrn 32625 fpwrelmap 32745 eulerpartlemgvv 34379 fineqvrep 35110 satfvsucsuc 35371 bj-opelopabid 37189 phpreu 37612 poimirlem26 37654 vvdifopab 38262 brabidgaw 38367 diclspsn 41197 areaquad 43233 sprsymrelf 47487 | 
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