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Theorem ineleq 35478
Description: Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)
Assertion
Ref Expression
ineleq (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
Distinct variable groups:   𝑧,𝐵   𝑧,𝐶   𝑧,𝐷   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem ineleq
StepHypRef Expression
1 orcom 866 . . . . 5 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦))
2 df-or 844 . . . . 5 (((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦) ↔ (¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦))
3 neq0 4312 . . . . . . . 8 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐶𝐷))
4 elin 4172 . . . . . . . . 9 (𝑧 ∈ (𝐶𝐷) ↔ (𝑧𝐶𝑧𝐷))
54exbii 1841 . . . . . . . 8 (∃𝑧 𝑧 ∈ (𝐶𝐷) ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
63, 5bitri 276 . . . . . . 7 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
76imbi1i 351 . . . . . 6 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
8 19.23v 1936 . . . . . 6 (∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
97, 8bitr4i 279 . . . . 5 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
101, 2, 93bitri 298 . . . 4 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1110ralbii 3169 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
12 ralcom4 3239 . . 3 (∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1311, 12bitri 276 . 2 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1413ralbii 3169 1 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  wal 1528   = wceq 1530  wex 1773  wcel 2107  wral 3142  cin 3938  c0 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-v 3501  df-dif 3942  df-in 3946  df-nul 4295
This theorem is referenced by:  inecmo  35479
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