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Theorem rexiunxp 4478
 Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 4480, 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 4475 . . . . . 6 (𝑥 𝑦𝐴 ({𝑦} × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21anbi1i 431 . . . . 5 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
3 19.41vv 1783 . . . . 5 (∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
42, 3bitr4i 176 . . . 4 ((𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
54exbii 1496 . . 3 (∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
6 exrot3 1580 . . . 4 (∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
7 anass 381 . . . . . . 7 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
87exbii 1496 . . . . . 6 (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
9 vex 2560 . . . . . . . 8 𝑦 ∈ V
10 vex 2560 . . . . . . . 8 𝑧 ∈ V
119, 10opex 3966 . . . . . . 7 𝑦, 𝑧⟩ ∈ V
12 ralxp.1 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
1312anbi2d 437 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓)))
1411, 13ceqsexv 2593 . . . . . 6 (∃𝑥(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
158, 14bitri 173 . . . . 5 (∃𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
16152exbii 1497 . . . 4 (∃𝑦𝑧𝑥((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
176, 16bitri 173 . . 3 (∃𝑥𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
185, 17bitri 173 . 2 (∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
19 df-rex 2312 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑥(𝑥 𝑦𝐴 ({𝑦} × 𝐵) ∧ 𝜑))
20 r2ex 2344 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ∃𝑦𝑧((𝑦𝐴𝑧𝐵) ∧ 𝜓))
2118, 19, 203bitr4i 201 1 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  {csn 3375  ⟨cop 3378  ∪ ciun 3657   × cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-opab 3819  df-xp 4351  df-rel 4352 This theorem is referenced by:  rexxp  4480
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