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Theorem ineleq 34461
 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 859 . . . . 5 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦))
2 df-or 837 . . . . 5 (((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦) ↔ (¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦))
3 neq0 4078 . . . . . . . 8 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐶𝐷))
4 elin 3947 . . . . . . . . 9 (𝑧 ∈ (𝐶𝐷) ↔ (𝑧𝐶𝑧𝐷))
54exbii 1924 . . . . . . . 8 (∃𝑧 𝑧 ∈ (𝐶𝐷) ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
63, 5bitri 264 . . . . . . 7 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
76imbi1i 338 . . . . . 6 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
8 19.23v 2023 . . . . . 6 (∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
97, 8bitr4i 267 . . . . 5 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
101, 2, 93bitri 286 . . . 4 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1110ralbii 3129 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
12 ralcom4 3376 . . 3 (∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1311, 12bitri 264 . 2 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1413ralbii 3129 1 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 836  ∀wal 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061   ∩ cin 3722  ∅c0 4063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064 This theorem is referenced by:  inecmo  34462
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