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Mirrors > Home > MPE Home > Th. List > cbvopab1v | Structured version Visualization version GIF version |
Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) Reduce axiom usage. (Revised by GG, 17-Nov-2024.) |
Ref | Expression |
---|---|
cbvopab1v.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvopab1v | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4878 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 〈𝑥, 𝑦〉 = 〈𝑧, 𝑦〉) | |
2 | 1 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑤 = 〈𝑥, 𝑦〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
3 | cbvopab1v.1 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜓))) |
5 | 4 | exbidv 1919 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜓))) |
6 | 5 | cbvexvw 2034 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜓)) |
7 | 6 | abbii 2807 | . 2 ⊢ {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜓)} |
8 | df-opab 5211 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
9 | df-opab 5211 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜓)} | |
10 | 7, 8, 9 | 3eqtr4i 2773 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 {cab 2712 〈cop 4637 {copab 5210 |
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-opab 5211 |
This theorem is referenced by: cbvmptv 5261 cbvmptvw2 36217 |
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