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| Description: The law of concretion in terms of substitutions. Version of opelopabsb 5534 with set variables. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Remove unnecessary commutation. (Revised by SN, 1-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| vopelopabsb | ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqcom 2743 | . . . . 5 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ 〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉) | |
| 2 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | vex 3483 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | opth 5480 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) | 
| 5 | 1, 4 | bitri 275 | . . . 4 ⊢ (〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) | 
| 6 | 5 | anbi1i 624 | . . 3 ⊢ ((〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) | 
| 7 | 6 | 2exbii 1848 | . 2 ⊢ (∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) | 
| 8 | elopab 5531 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(〈𝑧, 𝑤〉 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
| 9 | 2sb5 2277 | . 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) | |
| 10 | 7, 8, 9 | 3bitr4i 303 | 1 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 [wsb 2063 ∈ 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-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: opelopabsb 5534 inopab 5838 difopab 5839 cnvopabOLD 6157 isarep1 6655 brabsb2 38864 | 
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