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Theorem raaan 3369
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 3368 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
43pm5.74i 178 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓)) ↔ (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
5 ralm 3367 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓)) ↔ ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
6 jcab 568 . . 3 ((∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) ↔ ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ∧ (∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓)))
7 ralm 3367 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
8 eleq1 2145 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98cbvexv 1838 . . . . . 6 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
109imbi1i 236 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ (∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓))
11 ralm 3367 . . . . 5 ((∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
1210, 11bitri 182 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
137, 12anbi12i 448 . . 3 (((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ∧ (∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
146, 13bitri 182 . 2 ((∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
154, 5, 143bitr3i 208 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wnf 1390  wex 1422  wcel 1434  wral 2353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358
This theorem is referenced by:  raaanv  3370
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