Theorem raliunxp 4752

Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4754, 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 4750	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	imbi1i 237	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
3		19.23vv 1877	. . . . 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 186	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
5	4	albii 1463	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
6		alrot3 1478	. . . 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 261	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
8	7	albii 1463	. . . . . 6 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
9		vex 2733	. . . . . . . 8 ⊢ 𝑦 ∈ V
10		vex 2733	. . . . . . . 8 ⊢ 𝑧 ∈ V
11	9, 10	opex 4214	. . . . . . 7 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
12		ralxp.1	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
13	12	imbi2d 229	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓)))
14	11, 13	ceqsalv 2760	. . . . . 6 ⊢ (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
15	8, 14	bitri 183	. . . . 5 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
16	15	2albii 1464	. . . 4 ⊢ (∀𝑦∀𝑧∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
17	6, 16	bitri 183	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
18	5, 17	bitri 183	. 2 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
19		df-ral 2453	. 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑))
20		r2al 2489	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
21	18, 19, 20	3bitr4i 211	1 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  {csn 3583  ⟨cop 3586  ∪ ciun 3873   × cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-opab 4051  df-xp 4617  df-rel 4618

This theorem is referenced by:  ralxp  4754  fmpox  6179