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Mirrors > Home > MPE Home > Th. List > cbvoprab3 | Structured version Visualization version GIF version |
Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
cbvoprab3.1 | ⊢ Ⅎ𝑤𝜑 |
cbvoprab3.2 | ⊢ Ⅎ𝑧𝜓 |
cbvoprab3.3 | ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab3 | ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩ | |
2 | cbvoprab3.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
4 | 3 | nfex 2309 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
5 | 4 | nfex 2309 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
6 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩ | |
7 | cbvoprab3.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜓 | |
8 | 6, 7 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) |
9 | 8 | nfex 2309 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) |
10 | 9 | nfex 2309 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) |
11 | cbvoprab3.3 | . . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) | |
12 | 11 | anbi2d 628 | . . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))) |
13 | 12 | 2exbidv 1919 | . . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))) |
14 | 5, 10, 13 | cbvopab2 5215 | . 2 ⊢ {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} |
15 | dfoprab2 7459 | . 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} | |
16 | dfoprab2 7459 | . 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} | |
17 | 14, 15, 16 | 3eqtr4i 2762 | 1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 ⟨cop 4626 {copab 5200 {coprab 7402 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-opab 5201 df-oprab 7405 |
This theorem is referenced by: cbvoprab3v 7493 tposoprab 8242 erovlem 8802 |
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