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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 631 | . . . . 5 ⊢ (𝜑 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
5 | 2, 4 | exbid 2223 | . . . 4 ⊢ (𝜑 → (∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
6 | 1, 5 | exbid 2223 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒))) |
7 | 6 | abbidv 2862 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒)}) |
8 | df-opab 5093 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
9 | df-opab 5093 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜒} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜒)} | |
10 | 7, 8, 9 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 {cab 2776 〈cop 4531 {copab 5092 |
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 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-opab 5093 |
This theorem is referenced by: opabbidv 5096 mpteq12df 5112 mpteq12f 5113 feqmptdf 6710 fnoprabg 7254 sprsymrelfo 44014 |
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