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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 1843 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab3.1 | . . . . . 6 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑤(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfex 2154 | . . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | 4 | nfex 2154 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑧 𝑣 = 〈𝑥, 𝑦〉 | |
7 | cbvoprab3.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜓 | |
8 | 6, 7 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑧(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
9 | 8 | nfex 2154 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
10 | 9 | nfex 2154 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓) |
11 | cbvoprab3.3 | . . . . 5 ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) | |
12 | 11 | anbi2d 740 | . . . 4 ⊢ (𝑧 = 𝑤 → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
13 | 12 | 2exbidv 1852 | . . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
14 | 5, 10, 13 | cbvopab2 4724 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} |
15 | dfoprab2 6701 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
16 | dfoprab2 6701 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} = {〈𝑣, 𝑤〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
17 | 14, 15, 16 | 3eqtr4i 2654 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 Ⅎwnf 1708 〈cop 4183 {copab 4712 {coprab 6651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-oprab 6654 |
This theorem is referenced by: cbvoprab3v 6732 tposoprab 7388 erovlem 7843 |
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