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Mirrors > Home > MPE Home > Th. List > opabbid | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
opabbid.1 | ⊢ Ⅎ𝑥𝜑 |
opabbid.2 | ⊢ Ⅎ𝑦𝜑 |
opabbid.3 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opabbid | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabbid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | opabbid.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | opabbid.3 | . . . . . 6 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 3 | anbi2d 632 | . . . . 5 ⊢ (𝜑 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
5 | 2, 4 | exbid 2221 | . . . 4 ⊢ (𝜑 → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
6 | 1, 5 | exbid 2221 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
7 | 6 | abbidv 2807 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒)}) |
8 | df-opab 5121 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
9 | df-opab 5121 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜒} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒)} | |
10 | 7, 8, 9 | 3eqtr4g 2803 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 Ⅎwnf 1791 {cab 2714 〈cop 4552 {copab 5120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-opab 5121 |
This theorem is referenced by: mpteq12df 5142 mpteq12f 5143 feqmptdf 6787 fnoprabg 7338 sprsymrelfo 44630 |
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