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Theorem raaan 3392
 Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
Hypotheses
Ref Expression
raaan.1 𝑦𝜑
raaan.2 𝑥𝜓
Assertion
Ref Expression
raaan (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem raaan
StepHypRef Expression
1 raaan.1 . . . 4 𝑦𝜑
2 raaan.2 . . . 4 𝑥𝜓
31, 2raaanlem 3391 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
43pm5.74i 179 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓)) ↔ (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
5 ralm 3390 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓)) ↔ ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
6 jcab 571 . . 3 ((∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) ↔ ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ∧ (∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓)))
7 ralm 3390 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
8 eleq1 2151 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98cbvexv 1844 . . . . . 6 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
109imbi1i 237 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ (∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓))
11 ralm 3390 . . . . 5 ((∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
1210, 11bitri 183 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
137, 12anbi12i 449 . . 3 (((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ∧ (∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
146, 13bitri 183 . 2 ((∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
154, 5, 143bitr3i 209 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  Ⅎwnf 1395  ∃wex 1427   ∈ wcel 1439  ∀wral 2360 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365 This theorem is referenced by:  raaanv  3393
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