Theorem raliunxp 4898

Description: Write a double restricted quantification as one universal quantifier. In this version of ralxp 4900, 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 4896	. . . . . 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 1933	. . . . 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 1519	. . 3 ⊢ (∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑))
6		alrot3 1534	. . . 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 1519	. . . . . 6 ⊢ (∀𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝜑) ↔ ∀𝑥(𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → 𝜑)))
9		vex 2818	. . . . . . . 8 ⊢ 𝑦 ∈ V
10		vex 2818	. . . . . . . 8 ⊢ 𝑧 ∈ V
11	9, 10	opex 4347	. . . . . . 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 2846	. . . . . 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 1520	. . . 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 2527	. 2 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) → 𝜑))
20		r2al 2563	. 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)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  {csn 3691  ⟨cop 3694  ∪ ciun 3993   × cxp 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-iun 3995  df-opab 4174  df-xp 4757  df-rel 4758

This theorem is referenced by:  ralxp  4900  fmpox  6398