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Theorem ineleq 34480
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 896 . . . . 5 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦))
2 df-or 874 . . . . 5 (((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦) ↔ (¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦))
3 neq0 4094 . . . . . . . 8 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐶𝐷))
4 elin 3958 . . . . . . . . 9 (𝑧 ∈ (𝐶𝐷) ↔ (𝑧𝐶𝑧𝐷))
54exbii 1943 . . . . . . . 8 (∃𝑧 𝑧 ∈ (𝐶𝐷) ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
63, 5bitri 266 . . . . . . 7 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
76imbi1i 340 . . . . . 6 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
8 19.23v 2037 . . . . . 6 (∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
97, 8bitr4i 269 . . . . 5 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
101, 2, 93bitri 288 . . . 4 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1110ralbii 3127 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
12 ralcom4 3377 . . 3 (∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1311, 12bitri 266 . 2 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1413ralbii 3127 1 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  wal 1650   = wceq 1652  wex 1874  wcel 2155  wral 3055  cin 3731  c0 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-dif 3735  df-in 3739  df-nul 4080
This theorem is referenced by:  inecmo  34481
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