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Mirrors > Home > MPE Home > Th. List > raaan | Structured version Visualization version GIF version |
Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.) |
Ref | Expression |
---|---|
raaan.1 | ⊢ Ⅎ𝑦𝜑 |
raaan.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
raaan | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rzal 4217 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓)) | |
2 | rzal 4217 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | |
3 | rzal 4217 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝜓) | |
4 | pm5.1 938 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) | |
5 | 1, 2, 3, 4 | syl12anc 1475 | . 2 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
6 | raaan.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
7 | 6 | r19.28z 4207 | . . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
8 | 7 | ralbidv 3124 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
9 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
10 | raaan.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
11 | 9, 10 | nfral 3083 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓 |
12 | 11 | r19.27z 4214 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
13 | 8, 12 | bitrd 268 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
14 | 5, 13 | pm2.61ine 3015 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1632 Ⅎwnf 1857 ≠ wne 2932 ∀wral 3050 ∅c0 4058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-v 3342 df-dif 3718 df-nul 4059 |
This theorem is referenced by: raaanv 4227 |
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