Theorem raliunxp 4921

Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4923, 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 4919	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	imbi1i 238	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
3		19.23vv 1937	. . . . 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 187	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
5	4	albii 1523	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
6		alrot3 1538	. . . 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 263	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
8	7	albii 1523	. . . . . 6 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
9		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
10		vex 2824	. . . . . . . 8 ⊢ 𝑧 ∈ V
11	9, 10	opex 4369	. . . . . . 7 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
12		ralxp.1	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
13	12	imbi2d 230	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓)))
14	11, 13	ceqsalv 2852	. . . . . 6 ⊢ (∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
15	8, 14	bitri 184	. . . . 5 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
16	15	2albii 1524	. . . 4 ⊢ (∀𝑦∀𝑧∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
17	6, 16	bitri 184	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
18	5, 17	bitri 184	. 2 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
19		df-ral 2533	. 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑))
20		r2al 2569	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓 ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜓))
21	18, 19, 20	3bitr4i 212	1 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {csn 3709  ⟨cop 3712  ∪ ciun 4012   × cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-iun 4014  df-opab 4193  df-xp 4780  df-rel 4781

This theorem is used by:  ralxp  4923  fmpox  6436