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Mirrors > Home > MPE Home > Th. List > cbvopab2 | Structured version Visualization version GIF version |
Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
cbvopab2.1 | ⊢ Ⅎ𝑧𝜑 |
cbvopab2.2 | ⊢ Ⅎ𝑦𝜓 |
cbvopab2.3 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopab2 | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉 | |
2 | cbvopab2.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfan 1891 | . . . . 5 ⊢ Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑦 𝑤 = 〈𝑥, 𝑧〉 | |
5 | cbvopab2.2 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
6 | 4, 5 | nfan 1891 | . . . . 5 ⊢ Ⅎ𝑦(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓) |
7 | opeq2 4796 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑥, 𝑧〉) | |
8 | 7 | eqeq2d 2829 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑥, 𝑧〉)) |
9 | cbvopab2.3 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | anbi12d 630 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓))) |
11 | 3, 6, 10 | cbvexv1 2353 | . . . 4 ⊢ (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
12 | 11 | exbii 1839 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)) |
13 | 12 | abbii 2883 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} |
14 | df-opab 5120 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
15 | df-opab 5120 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜓)} | |
16 | 13, 14, 15 | 3eqtr4i 2851 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 {cab 2796 〈cop 4563 {copab 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 |
This theorem is referenced by: cbvoprab3 7234 |
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