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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 1909 | . . . . 5 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩ | |
2 | cbvopab.1 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
4 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩ | |
5 | cbvopab.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
6 | 4, 5 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
7 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑧, 𝑤⟩ | |
8 | cbvopab.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
9 | 7, 8 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓) |
10 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑦 𝑣 = ⟨𝑧, 𝑤⟩ | |
11 | cbvopab.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 10, 11 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓) |
13 | opeq12 4870 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩) | |
14 | 13 | eqeq2d 2737 | . . . . 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 630 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))) |
17 | 3, 6, 9, 12, 16 | cbvex2v 2334 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)) |
18 | 17 | abbii 2796 | . 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)} |
19 | df-opab 5204 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} | |
20 | df-opab 5204 | . 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)} | |
21 | 18, 19, 20 | 3eqtr4i 2764 | 1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 {cab 2703 ⟨cop 4629 {copab 5203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 |
This theorem is referenced by: cbvopabvOLD 5215 dfrel4 6183 bj-opabco 36575 aomclem8 42363 |
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