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Theorem ineleq 38335
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 870 . . . . 5 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦))
2 df-or 848 . . . . 5 (((𝐶𝐷) = ∅ ∨ 𝑥 = 𝑦) ↔ (¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦))
3 neq0 4357 . . . . . . . 8 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐶𝐷))
4 elin 3978 . . . . . . . . 9 (𝑧 ∈ (𝐶𝐷) ↔ (𝑧𝐶𝑧𝐷))
54exbii 1844 . . . . . . . 8 (∃𝑧 𝑧 ∈ (𝐶𝐷) ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
63, 5bitri 275 . . . . . . 7 (¬ (𝐶𝐷) = ∅ ↔ ∃𝑧(𝑧𝐶𝑧𝐷))
76imbi1i 349 . . . . . 6 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
8 19.23v 1939 . . . . . 6 (∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
97, 8bitr4i 278 . . . . 5 ((¬ (𝐶𝐷) = ∅ → 𝑥 = 𝑦) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
101, 2, 93bitri 297 . . . 4 ((𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1110ralbii 3090 . . 3 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
12 ralcom4 3283 . . 3 (∀𝑦𝐵𝑧((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1311, 12bitri 275 . 2 (∀𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
1413ralbii 3090 1 (∀𝑥𝐴𝑦𝐵 (𝑥 = 𝑦 ∨ (𝐶𝐷) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐵 ((𝑧𝐶𝑧𝐷) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  wal 1534   = wceq 1536  wex 1775  wcel 2105  wral 3058  cin 3961  c0 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-v 3479  df-dif 3965  df-in 3969  df-nul 4339
This theorem is referenced by:  inecmo  38336
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