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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 4336 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓)) | |
2 | rzal 4336 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | |
3 | rzal 4336 | . . 3 ⊢ (𝐴 = ∅ → ∀𝑦 ∈ 𝐴 𝜓) | |
4 | pm5.1 812 | . . 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 824 | . 2 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
6 | raaan.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
7 | 6 | r19.28z 4326 | . . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
8 | 7 | ralbidv 3147 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
9 | nfcv 2932 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
10 | raaan.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
11 | 9, 10 | nfral 3174 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓 |
12 | 11 | r19.27z 4333 | . . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
13 | 8, 12 | bitrd 271 | . 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))) |
14 | 5, 13 | pm2.61ine 3051 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 Ⅎwnf 1746 ≠ wne 2967 ∀wral 3088 ∅c0 4178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-dif 3832 df-nul 4179 |
This theorem is referenced by: (None) |
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