Theorem raliunxp 5824

Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 5826, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypothesis

Ref	Expression
ralxp.1	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
raliunxp	⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝐵(𝑦)

Proof of Theorem raliunxp

Step	Hyp	Ref	Expression
1		eliunxp 5822	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	imbi1i 349	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
3		19.23vv 1943	. . . . 5 ⊢ (∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
4	2, 3	bitr4i 278	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
5	4	albii 1819	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
6		alrot3 2161	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
7		impexp 450	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
8	7	albii 1819	. . . . . 6 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
9		opex 5444	. . . . . . 7 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
10		ralxp.1	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
11	10	imbi2d 340	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓)))
12	9, 11	ceqsalv 3505	. . . . . 6 ⊢ (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
13	8, 12	bitri 275	. . . . 5 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
14	13	2albii 1820	. . . 4 ⊢ (∀𝑦∀𝑧∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
15	6, 14	bitri 275	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
16	5, 15	bitri 275	. 2 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
17		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑))
18		r2al 3181	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
19	16, 17, 18	3bitr4i 303	1 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3052  {csn 4606  ⟨cop 4612  ∪ ciun 4972   × cxp 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-iun 4974  df-opab 5187  df-xp 5665  df-rel 5666

This theorem is referenced by:  rexiunxp  5825  ralxp  5826  fmpox  8071  ovmptss  8097  filnetlem4  36404