Theorem rexiunxp 4676

Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 4678, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)

Hypothesis

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

Assertion

Ref	Expression
rexiunxp	⊢ (∃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 rexiunxp

Step	Hyp	Ref	Expression
1		eliunxp 4673	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	anbi1i 453	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
3		19.41vv 1875	. . . . 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	exbii 1584	. . 3 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
6		exrot3 1668	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ ∃𝑦∃𝑧∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
7		anass 398	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑)))
8	7	exbii 1584	. . . . . 6 ⊢ (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ ∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑)))
9		vex 2684	. . . . . . . 8 ⊢ 𝑦 ∈ V
10		vex 2684	. . . . . . . 8 ⊢ 𝑧 ∈ V
11	9, 10	opex 4146	. . . . . . 7 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
12		ralxp.1	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
13	12	anbi2d 459	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜓)))
14	11, 13	ceqsexv 2720	. . . . . 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	2exbii 1585	. . . 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-rex 2420	. 2 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ∧ 𝜑))
20		r2ex 2453	. 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   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415  {csn 3522  ⟨cop 3525  ∪ ciun 3808   × cxp 4532

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-iun 3810  df-opab 3985  df-xp 4540  df-rel 4541

This theorem is referenced by:  rexxp  4678