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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 1906 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab3.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1891 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfex 2334 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | 4 | nfex 2334 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑧 𝑣 = 〈𝑥, 𝑦〉 | |
7 | cbvoprab3.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜓 | |
8 | 6, 7 | nfan 1891 | . . . . 5 ⊢ Ⅎ𝑧(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
9 | 8 | nfex 2334 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
10 | 9 | nfex 2334 | . . 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 1916 | . . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
14 | 5, 10, 13 | cbvopab2 5132 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} |
15 | dfoprab2 7201 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
16 | dfoprab2 7201 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
17 | 14, 15, 16 | 3eqtr4i 2851 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 〈cop 4563 {copab 5119 {coprab 7146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-oprab 7149 |
This theorem is referenced by: cbvoprab3v 7235 tposoprab 7917 erovlem 8382 |
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