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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 1822 | . . 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 5534 | . . . 4 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒)) |
6 | annim 402 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜒) ↔ ¬ (𝜑 → 𝜒)) | |
7 | 6 | 2exbii 1843 | . . . 4 ⊢ (∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜒) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒)) |
8 | 5, 7 | bitri 274 | . . 3 ⊢ (∃𝑜 ∈ 𝑂 ¬ 𝜓 ↔ ∃𝑥∃𝑦 ¬ (𝜑 → 𝜒)) |
9 | rexnal 3097 | . . 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 394 ∀wal 1531 = wceq 1533 ∃wex 1773 ∀wral 3058 ∃wrex 3067 ⟨cop 4638 {copab 5214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-opab 5215 |
This theorem is referenced by: (None) |
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