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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 4874 | . . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 〈𝑥, 𝑦〉 = 〈𝑤, 𝑣〉) | |
| 2 | 1 | opeq1d 4878 | . . . . . . 7 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈〈𝑤, 𝑣〉, 𝑧〉) | 
| 3 | 2 | eqeq2d 2747 | . . . . . 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 1920 | . . . 4 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) ↔ ∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓))) | 
| 7 | 6 | cbvex2vw 2039 | . . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) ↔ ∃𝑤∃𝑣∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓)) | 
| 8 | 7 | abbii 2808 | . 2 ⊢ {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓)} | 
| 9 | df-oprab 7436 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑢 ∣ ∃𝑥∃𝑦∃𝑧(𝑢 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
| 10 | df-oprab 7436 | . 2 ⊢ {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} = {𝑢 ∣ ∃𝑤∃𝑣∃𝑧(𝑢 = 〈〈𝑤, 𝑣〉, 𝑧〉 ∧ 𝜓)} | |
| 11 | 8, 9, 10 | 3eqtr4i 2774 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑤, 𝑣〉, 𝑧〉 ∣ 𝜓} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 {cab 2713 〈cop 4631 {coprab 7433 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-oprab 7436 | 
| This theorem is referenced by: cbvmpov 7529 cpnnen 16266 cbvmpovw2 36244 | 
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