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Theorem raaan 3327
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 3326 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
43pm5.74i 169 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓)) ↔ (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
5 ralm 3325 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓)) ↔ ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
6 jcab 535 . . 3 ((∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) ↔ ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ∧ (∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓)))
7 ralm 3325 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
8 eleq1 2100 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98cbvexv 1795 . . . . . 6 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
109imbi1i 227 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ (∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓))
11 ralm 3325 . . . . 5 ((∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
1210, 11bitri 173 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
137, 12anbi12i 433 . . 3 (((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ∧ (∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
146, 13bitri 173 . 2 ((∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
154, 5, 143bitr3i 199 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wnf 1349  wex 1381  wcel 1393  wral 2306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311
This theorem is referenced by:  raaanv  3328
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