Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ineleq Structured version   Visualization version   GIF version

Theorem ineleq 36074
 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 867 . . . . 5 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦))
2 df-or 845 . . . . 5 (((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦) ↔ (¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦))
3 neq0 4246 . . . . . . . 8 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐶𝐷))
4 elin 3876 . . . . . . . . 9 (𝑧 ∈ (𝐶𝐷) ↔ (𝑧𝐶𝑧𝐷))
54exbii 1849 . . . . . . . 8 (∃𝑧 𝑧 ∈ (𝐶𝐷) ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
63, 5bitri 278 . . . . . . 7 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
76imbi1i 353 . . . . . 6 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
8 19.23v 1943 . . . . . 6 (∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
97, 8bitr4i 281 . . . . 5 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
101, 2, 93bitri 300 . . . 4 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1110ralbii 3097 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
12 ralcom4 3162 . . 3 (∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1311, 12bitri 278 . 2 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1413ralbii 3097 1 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070   ∩ cin 3859  ∅c0 4227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-dif 3863  df-in 3867  df-nul 4228 This theorem is referenced by:  inecmo  36075
 Copyright terms: Public domain W3C validator