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Mirrors > Home > MPE Home > Th. List > cbvoprab12 | Structured version Visualization version GIF version |
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
cbvoprab12.1 | ⊢ Ⅎ𝑤𝜑 |
cbvoprab12.2 | ⊢ Ⅎ𝑣𝜑 |
cbvoprab12.3 | ⊢ Ⅎ𝑥𝜓 |
cbvoprab12.4 | ⊢ Ⅎ𝑦𝜓 |
cbvoprab12.5 | ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab12 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑤 𝑢 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab12.1 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
3 | 1, 2 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑤(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑣 𝑢 = 〈𝑥, 𝑦〉 | |
5 | cbvoprab12.2 | . . . . 5 ⊢ Ⅎ𝑣𝜑 | |
6 | 4, 5 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑣(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑥 𝑢 = 〈𝑤, 𝑣〉 | |
8 | cbvoprab12.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
9 | 7, 8 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑥(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓) |
10 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑦 𝑢 = 〈𝑤, 𝑣〉 | |
11 | cbvoprab12.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 10, 11 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑦(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓) |
13 | opeq12 4798 | . . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑣〉) | |
14 | 13 | eqeq2d 2832 | . . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝑢 = 〈𝑥, 𝑦〉 ↔ 𝑢 = 〈𝑤, 𝑣〉)) |
15 | cbvoprab12.5 | . . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) | |
16 | 14, 15 | anbi12d 632 | . . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ((𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓))) |
17 | 3, 6, 9, 12, 16 | cbvex2v 2361 | . . 3 ⊢ (∃𝑥∃𝑦(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑤∃𝑣(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓)) |
18 | 17 | opabbii 5125 | . 2 ⊢ {〈𝑢, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑢, 𝑧〉 ∣ ∃𝑤∃𝑣(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓)} |
19 | dfoprab2 7206 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑢, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑢 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
20 | dfoprab2 7206 | . 2 ⊢ {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} = {〈𝑢, 𝑧〉 ∣ ∃𝑤∃𝑣(𝑢 = 〈𝑤, 𝑣〉 ∧ 𝜓)} | |
21 | 18, 19, 20 | 3eqtr4i 2854 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 Ⅎwnf 1780 〈cop 4566 {copab 5120 {coprab 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5121 df-oprab 7154 |
This theorem is referenced by: cbvoprab12v 7238 cbvmpox 7241 dfoprab4f 7748 fmpox 7759 tposoprab 7922 f1od2 30451 cbvmpox2 44378 |
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