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Theorem 2ralor 3138
Description: Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
Assertion
Ref Expression
2ralor (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2ralor
StepHypRef Expression
1 rexnal 3024 . . . 4 (∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)
2 rexnal 3024 . . . 4 (∃𝑦𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦𝐵 𝜓)
31, 2anbi12i 733 . . 3 ((∃𝑥𝐴 ¬ 𝜑 ∧ ∃𝑦𝐵 ¬ 𝜓) ↔ (¬ ∀𝑥𝐴 𝜑 ∧ ¬ ∀𝑦𝐵 𝜓))
4 ioran 510 . . . . . . 7 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
54rexbii 3070 . . . . . 6 (∃𝑦𝐵 ¬ (𝜑𝜓) ↔ ∃𝑦𝐵𝜑 ∧ ¬ 𝜓))
6 rexnal 3024 . . . . . 6 (∃𝑦𝐵 ¬ (𝜑𝜓) ↔ ¬ ∀𝑦𝐵 (𝜑𝜓))
75, 6bitr3i 266 . . . . 5 (∃𝑦𝐵𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦𝐵 (𝜑𝜓))
87rexbii 3070 . . . 4 (∃𝑥𝐴𝑦𝐵𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓))
9 reeanv 3136 . . . 4 (∃𝑥𝐴𝑦𝐵𝜑 ∧ ¬ 𝜓) ↔ (∃𝑥𝐴 ¬ 𝜑 ∧ ∃𝑦𝐵 ¬ 𝜓))
10 rexnal 3024 . . . 4 (∃𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓) ↔ ¬ ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
118, 9, 103bitr3ri 291 . . 3 (¬ ∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 ¬ 𝜑 ∧ ∃𝑦𝐵 ¬ 𝜓))
12 ioran 510 . . 3 (¬ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓) ↔ (¬ ∀𝑥𝐴 𝜑 ∧ ¬ ∀𝑦𝐵 𝜓))
133, 11, 123bitr4i 292 . 2 (¬ ∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
1413con4bii 310 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-ral 2946  df-rex 2947
This theorem is referenced by:  ispridl2  33967
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