Theorem ineleq 39063

Description: Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)

Assertion

Ref	Expression
ineleq	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 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

Step	Hyp	Ref	Expression
1		orcom 884	. . . . 5 ⊢ ((𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ((𝐶 ∩ 𝐷) = ∅ ∨ 𝑥 = 𝑦))
2		df-or 862	. . . . 5 ⊢ (((𝐶 ∩ 𝐷) = ∅ ∨ 𝑥 = 𝑦) ↔ (¬ (𝐶 ∩ 𝐷) = ∅ → 𝑥 = 𝑦))
3		neq0 4306	. . . . . . . 8 ⊢ (¬ (𝐶 ∩ 𝐷) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐶 ∩ 𝐷))
4		elin 3922	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐶 ∩ 𝐷) ↔ (𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷))
5	4	exbii 1881	. . . . . . . 8 ⊢ (∃𝑧 𝑧 ∈ (𝐶 ∩ 𝐷) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷))
6	3, 5	bitri 278	. . . . . . 7 ⊢ (¬ (𝐶 ∩ 𝐷) = ∅ ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷))
7	6	imbi1i 352	. . . . . 6 ⊢ ((¬ (𝐶 ∩ 𝐷) = ∅ → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
8		19.23v 1975	. . . . . 6 ⊢ (∀𝑧((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
9	7, 8	bitr4i 281	. . . . 5 ⊢ ((¬ (𝐶 ∩ 𝐷) = ∅ → 𝑥 = 𝑦) ↔ ∀𝑧((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
10	1, 2, 9	3bitri 300	. . . 4 ⊢ ((𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑧((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
11	10	ralbii 3113	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
12		ralcom4 3293	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
13	11, 12	bitri 278	. 2 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
14	13	ralbii 3113	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081   ∩ cin 3905  ∅c0 4286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-dif 3909  df-in 3913  df-nul 4287

This theorem is used by:  inecmo  39064