Theorem raliunxp 5842

Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 5844, 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 5840	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	imbi1i 348	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
3		19.23vv 1938	. . . . 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 277	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
5	4	albii 1813	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
6		alrot3 2149	. . . 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 449	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
8	7	albii 1813	. . . . . 6 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
9		opex 5466	. . . . . . 7 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
10		ralxp.1	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
11	10	imbi2d 339	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓)))
12	9, 11	ceqsalv 3500	. . . . . 6 ⊢ (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
13	8, 12	bitri 274	. . . . 5 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
14	13	2albii 1814	. . . 4 ⊢ (∀𝑦∀𝑧∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
15	6, 14	bitri 274	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
16	5, 15	bitri 274	. 2 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
17		df-ral 3051	. 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑))
18		r2al 3184	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
19	16, 17, 18	3bitr4i 302	1 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3050  {csn 4630  ⟨cop 4636  ∪ ciun 4997   × cxp 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-iun 4999  df-opab 5212  df-xp 5684  df-rel 5685

This theorem is referenced by:  rexiunxp  5843  ralxp  5844  fmpox  8072  ovmptss  8098  filnetlem4  35996