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Mirrors > Home > MPE Home > Th. List > cbvoprab12v | 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, 8-Oct-2004.) |
Ref | Expression |
---|---|
cbvoprab12v.1 | ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvoprab12v | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq12 4880 | . . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑣〉) | |
2 | 1 | opeq1d 4884 | . . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈〈𝑤, 𝑣〉, 𝑧〉) |
3 | 2 | eqeq2d 2746 | . . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ 𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉)) |
4 | cbvoprab12v.1 | . . . . . 6 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | anbi12d 632 | . . . . 5 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → ((𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) ↔ (𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓))) |
6 | 5 | exbidv 1919 | . . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) ↔ ∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓))) |
7 | 6 | cbvex2vw 2038 | . . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) ↔ ∃𝑤∃𝑣∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓)) |
8 | 7 | abbii 2807 | . 2 ⊢ {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓)} |
9 | df-oprab 7435 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
10 | df-oprab 7435 | . 2 ⊢ {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓)} | |
11 | 8, 9, 10 | 3eqtr4i 2773 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 {cab 2712 〈cop 4637 {coprab 7432 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-oprab 7435 |
This theorem is referenced by: cbvmpov 7528 cpnnen 16262 cbvmpovw2 36225 |
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