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Theorem raaan 3437
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 3436 . . 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 3435 . 2 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓)) ↔ ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
6 jcab 575 . . 3 ((∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)) ↔ ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ∧ (∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓)))
7 ralm 3435 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
8 eleq1 2178 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98cbvexv 1870 . . . . . 6 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
109imbi1i 237 . . . . 5 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ (∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓))
11 ralm 3435 . . . . 5 ((∃𝑦 𝑦𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
1210, 11bitri 183 . . . 4 ((∃𝑥 𝑥𝐴 → ∀𝑦𝐴 𝜓) ↔ ∀𝑦𝐴 𝜓)
137, 12anbi12i 453 . . 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  wnf 1419  wex 1451  wcel 1463  wral 2391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396
This theorem is referenced by:  raaanv  3438
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