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| Mirrors > Home > MPE Home > Th. List > cbvopab | Structured version Visualization version GIF version | ||
| Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.) | 
| Ref | Expression | 
|---|---|
| cbvopab.1 | ⊢ Ⅎ𝑧𝜑 | 
| cbvopab.2 | ⊢ Ⅎ𝑤𝜑 | 
| cbvopab.3 | ⊢ Ⅎ𝑥𝜓 | 
| cbvopab.4 | ⊢ Ⅎ𝑦𝜓 | 
| cbvopab.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvopab | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑧 𝑣 = 〈𝑥, 𝑦〉 | |
| 2 | cbvopab.1 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
| 3 | 1, 2 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑧(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) | 
| 4 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
| 5 | cbvopab.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
| 6 | 4, 5 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) | 
| 7 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝑣 = 〈𝑧, 𝑤〉 | |
| 8 | cbvopab.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 9 | 7, 8 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓) | 
| 10 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝑣 = 〈𝑧, 𝑤〉 | |
| 11 | cbvopab.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
| 12 | 10, 11 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑦(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓) | 
| 13 | opeq12 4875 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑤〉) | |
| 14 | 13 | eqeq2d 2748 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = 〈𝑥, 𝑦〉 ↔ 𝑣 = 〈𝑧, 𝑤〉)) | 
| 15 | cbvopab.5 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
| 16 | 14, 15 | anbi12d 632 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓))) | 
| 17 | 3, 6, 9, 12, 16 | cbvex2v 2346 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)) | 
| 18 | 17 | abbii 2809 | . 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} | 
| 19 | df-opab 5206 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 20 | df-opab 5206 | . 2 ⊢ {〈𝑧, 𝑤〉 ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = 〈𝑧, 𝑤〉 ∧ 𝜓)} | |
| 21 | 18, 19, 20 | 3eqtr4i 2775 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑤〉 ∣ 𝜓} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 Ⅎwnf 1783 {cab 2714 〈cop 4632 {copab 5205 | 
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 | 
| This theorem is referenced by: dfrel4 6211 bj-opabco 37189 aomclem8 43073 | 
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