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Theorem rexiunxp 4506
Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 4508, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
ralxp.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
rexiunxp (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 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
StepHypRef Expression
1 eliunxp 4503 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21anbi1i 446 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
3 19.41vv 1825 . . . . 5 (∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
42, 3bitr4i 185 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
54exbii 1537 . . 3 (∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
6 exrot3 1621 . . . 4 (∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
7 anass 393 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
87exbii 1537 . . . . . 6 (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
9 vex 2605 . . . . . . . 8 𝑦 ∈ V
10 vex 2605 . . . . . . . 8 𝑧 ∈ V
119, 10opex 3992 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
12 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1312anbi2d 452 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓)))
1411, 13ceqsexv 2639 . . . . . 6 (∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
158, 14bitri 182 . . . . 5 (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
16152exbii 1538 . . . 4 (∃𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
176, 16bitri 182 . . 3 (∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
185, 17bitri 182 . 2 (∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
19 df-rex 2355 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑))
20 r2ex 2387 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
2118, 19, 203bitr4i 210 1 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  wrex 2350  {csn 3406  cop 3409   ciun 3686   × cxp 4369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-iun 3688  df-opab 3848  df-xp 4377  df-rel 4378
This theorem is referenced by:  rexxp  4508
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