Theorem ineleq 39023

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 883	. . . . 5 ⊢ ((𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ((𝐶 ∩ 𝐷) = ∅ ∨ 𝑥 = 𝑦))
2		df-or 861	. . . . 5 ⊢ (((𝐶 ∩ 𝐷) = ∅ ∨ 𝑥 = 𝑦) ↔ (¬ (𝐶 ∩ 𝐷) = ∅ → 𝑥 = 𝑦))
3		neq0 4306	. . . . . . . 8 ⊢ (¬ (𝐶 ∩ 𝐷) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐶 ∩ 𝐷))
4		elin 3921	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐶 ∩ 𝐷) ↔ (𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷))
5	4	exbii 1878	. . . . . . . 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 1972	. . . . . 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 3111	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
12		ralcom4 3291	. . 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 3111	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-dif 3908  df-in 3912  df-nul 4287

This theorem is referenced by:  inecmo  39024