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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralopabb | Structured version Visualization version GIF version |
Description: Restricted universal quantification over an ordered-pair class abstraction. (Contributed by RP, 25-Sep-2024.) |
Ref | Expression |
---|---|
ralopabb.o | ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
ralopabb.p | ⊢ (𝑜 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralopabb | ⊢ (∀𝑜 ∈ 𝑂 𝜓 ↔ ∀𝑥∀𝑦(𝜑 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nalexn 1830 | . . 3 ⊢ (¬ ∀𝑥∀𝑦(𝜑 → 𝜒) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒)) | |
2 | ralopabb.o | . . . . 5 ⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | ralopabb.p | . . . . . 6 ⊢ (𝑜 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜒)) | |
4 | 3 | notbid 317 | . . . . 5 ⊢ (𝑜 = 〈𝑥, 𝑦〉 → (¬ 𝜓 ↔ ¬ 𝜒)) |
5 | 2, 4 | rexopabb 5522 | . . . 4 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒)) |
6 | annim 404 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑 → 𝜒)) | |
7 | 6 | 2exbii 1851 | . . . 4 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒)) |
8 | 5, 7 | bitri 274 | . . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒)) |
9 | rexnal 3100 | . . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ¬ ∀𝑜 ∈ 𝑂 𝜓) | |
10 | 1, 8, 9 | 3bitr2ri 299 | . 2 ⊢ (¬ ∀𝑜 ∈ 𝑂 𝜓 ↔ ¬ ∀𝑥∀𝑦(𝜑 → 𝜒)) |
11 | 10 | con4bii 320 | 1 ⊢ (∀𝑜 ∈ 𝑂 𝜓 ↔ ∀𝑥∀𝑦(𝜑 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∃wex 1781 ∀wral 3061 ∃wrex 3070 〈cop 4629 {copab 5204 |
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 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
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-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5205 |
This theorem is referenced by: (None) |
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