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Mirrors > Home > MPE Home > Th. List > cbvoprab1 | Structured version Visualization version GIF version |
Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
cbvoprab1.1 | ⊢ Ⅎ𝑤𝜑 |
cbvoprab1.2 | ⊢ Ⅎ𝑥𝜓 |
cbvoprab1.3 | ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab1 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑤 𝑣 = 〈𝑥, 𝑦〉 | |
2 | cbvoprab1.1 | . . . . . 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 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑥 𝑣 = 〈𝑤, 𝑦〉 | |
6 | cbvoprab1.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
7 | 5, 6 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑥(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
8 | 7 | nfex 2154 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓) |
9 | opeq1 4402 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑦〉) | |
10 | 9 | eqeq2d 2632 | . . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑣 = 〈𝑥, 𝑦〉 ↔ 𝑣 = 〈𝑤, 𝑦〉)) |
11 | cbvoprab1.3 | . . . . . 6 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) | |
12 | 10, 11 | anbi12d 747 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓))) |
13 | 12 | exbidv 1850 | . . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓))) |
14 | 4, 8, 13 | cbvex 2272 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)) |
15 | 14 | opabbii 4717 | . 2 ⊢ {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} |
16 | dfoprab2 6701 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑣, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑣 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
17 | dfoprab2 6701 | . 2 ⊢ {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈𝑣, 𝑧〉 ∣ ∃𝑤∃𝑦(𝑣 = 〈𝑤, 𝑦〉 ∧ 𝜓)} | |
18 | 15, 16, 17 | 3eqtr4i 2654 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑦〉, 𝑧〉 ∣ 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: cbvmpt21 39278 |
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