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Mirrors > Home > MPE Home > Th. List > cbvopab1s | Structured version Visualization version GIF version |
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.) |
Ref | Expression |
---|---|
cbvopab1s | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
2 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 〈𝑧, 𝑦〉 | |
3 | nfs1v 2153 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
4 | 2, 3 | nfan 1902 | . . . . 5 ⊢ Ⅎ𝑥(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
5 | 4 | nfex 2317 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑) |
6 | opeq1 4828 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
7 | 6 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
8 | sbequ12 2243 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
9 | 7, 8 | anbi12d 631 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
10 | 9 | exbidv 1924 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑))) |
11 | 1, 5, 10 | cbvexv1 2338 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)) |
12 | 11 | abbii 2806 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} |
13 | df-opab 5166 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
14 | df-opab 5166 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ [𝑧 / 𝑥]𝜑)} | |
15 | 12, 13, 14 | 3eqtr4i 2774 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ [𝑧 / 𝑥]𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∃wex 1781 [wsb 2067 {cab 2713 〈cop 4590 {copab 5165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-opab 5166 |
This theorem is referenced by: (None) |
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