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Mirrors > Home > MPE Home > Th. List > cbvopab | Structured version Visualization version GIF version |
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
cbvopab.1 | ⊢ Ⅎ𝑧𝜑 |
cbvopab.2 | ⊢ Ⅎ𝑤𝜑 |
cbvopab.3 | ⊢ Ⅎ𝑥𝜓 |
cbvopab.4 | ⊢ Ⅎ𝑦𝜓 |
cbvopab.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopab | ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑧 𝑣 = ⟨𝑥, 𝑦⟩ | |
2 | cbvopab.1 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
4 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑤 𝑣 = ⟨𝑥, 𝑦⟩ | |
5 | cbvopab.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
6 | 4, 5 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
7 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑥 𝑣 = ⟨𝑧, 𝑤⟩ | |
8 | cbvopab.3 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
9 | 7, 8 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓) |
10 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑦 𝑣 = ⟨𝑧, 𝑤⟩ | |
11 | cbvopab.4 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
12 | 10, 11 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓) |
13 | opeq12 4876 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩) | |
14 | 13 | eqeq2d 2744 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑧, 𝑤⟩)) |
15 | cbvopab.5 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
16 | 14, 15 | anbi12d 632 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))) |
17 | 3, 6, 9, 12, 16 | cbvex2v 2341 | . . 3 ⊢ (∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)) |
18 | 17 | abbii 2803 | . 2 ⊢ {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)} |
19 | df-opab 5212 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} | |
20 | df-opab 5212 | . 2 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧∃𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)} | |
21 | 18, 19, 20 | 3eqtr4i 2771 | 1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 Ⅎwnf 1786 {cab 2710 ⟨cop 4635 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 |
This theorem is referenced by: cbvopabvOLD 5223 dfrel4 6191 bj-opabco 36069 aomclem8 41803 |
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