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Theorem 2ralor 3099
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 2989 . . . 4 (∃𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴 𝜑)
2 rexnal 2989 . . . 4 (∃𝑦𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦𝐵 𝜓)
31, 2anbi12i 732 . . 3 ((∃𝑥𝐴 ¬ 𝜑 ∧ ∃𝑦𝐵 ¬ 𝜓) ↔ (¬ ∀𝑥𝐴 𝜑 ∧ ¬ ∀𝑦𝐵 𝜓))
4 ioran 511 . . . . . . 7 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
54rexbii 3034 . . . . . 6 (∃𝑦𝐵 ¬ (𝜑𝜓) ↔ ∃𝑦𝐵𝜑 ∧ ¬ 𝜓))
6 rexnal 2989 . . . . . 6 (∃𝑦𝐵 ¬ (𝜑𝜓) ↔ ¬ ∀𝑦𝐵 (𝜑𝜓))
75, 6bitr3i 266 . . . . 5 (∃𝑦𝐵𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦𝐵 (𝜑𝜓))
87rexbii 3034 . . . 4 (∃𝑥𝐴𝑦𝐵𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓))
9 reeanv 3097 . . . 4 (∃𝑥𝐴𝑦𝐵𝜑 ∧ ¬ 𝜓) ↔ (∃𝑥𝐴 ¬ 𝜑 ∧ ∃𝑦𝐵 ¬ 𝜓))
10 rexnal 2989 . . . 4 (∃𝑥𝐴 ¬ ∀𝑦𝐵 (𝜑𝜓) ↔ ¬ ∀𝑥𝐴𝑦𝐵 (𝜑𝜓))
118, 9, 103bitr3ri 291 . . 3 (¬ ∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 ¬ 𝜑 ∧ ∃𝑦𝐵 ¬ 𝜓))
12 ioran 511 . . 3 (¬ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓) ↔ (¬ ∀𝑥𝐴 𝜑 ∧ ¬ ∀𝑦𝐵 𝜓))
133, 11, 123bitr4i 292 . 2 (¬ ∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ¬ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
1413con4bii 311 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384  wral 2907  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-ral 2912  df-rex 2913
This theorem is referenced by:  ispridl2  33469
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