Theorem rexiunxp 4746

Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 4748, 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 4743	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
2	1	anbi1i 454	. . . . 5 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
3		19.41vv 1891	. . . . 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 1593	. . 3 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑))
6		exrot3 1678	. . . 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 399	. . . . . . 7 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑)))
8	7	exbii 1593	. . . . . 6 ⊢ (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝜑) ↔ ∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑)))
9		vex 2729	. . . . . . . 8 ⊢ 𝑦 ∈ V
10		vex 2729	. . . . . . . 8 ⊢ 𝑧 ∈ V
11	9, 10	opex 4207	. . . . . . 7 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
12		ralxp.1	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
13	12	anbi2d 460	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ∧ 𝜓)))
14	11, 13	ceqsexv 2765	. . . . . 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 1594	. . . 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 2450	. 2 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ∧ 𝜑))
20		r2ex 2486	. 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 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  {csn 3576  ⟨cop 3579  ∪ ciun 3866   × cxp 4602

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-iun 3868  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by:  rexxp  4748