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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 1831 | . . 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 318 | . . . . 5 ⊢ (𝑜 = ⟨𝑥, 𝑦⟩ → (¬ 𝜓 ↔ ¬ 𝜒)) |
5 | 2, 4 | rexopabb 5529 | . . . 4 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒)) |
6 | annim 405 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑 → 𝜒)) | |
7 | 6 | 2exbii 1852 | . . . 4 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒)) |
8 | 5, 7 | bitri 275 | . . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒)) |
9 | rexnal 3101 | . . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ¬ ∀𝑜 ∈ 𝑂 𝜓) | |
10 | 1, 8, 9 | 3bitr2ri 300 | . 2 ⊢ (¬ ∀𝑜 ∈ 𝑂 𝜓 ↔ ¬ ∀𝑥∀𝑦(𝜑 → 𝜒)) |
11 | 10 | con4bii 321 | 1 ⊢ (∀𝑜 ∈ 𝑂 𝜓 ↔ ∀𝑥∀𝑦(𝜑 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∀wral 3062 ∃wrex 3071 ⟨cop 4635 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 |
This theorem is referenced by: (None) |
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