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Theorem rexiunxp 5850
Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 5852, 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 ralxp.1 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
21notbid 318 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
32raliunxp 5849 . . . 4 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴𝑧𝐵 ¬ 𝜓)
4 ralnex 3071 . . . . 5 (∀𝑧𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧𝐵 𝜓)
54ralbii 3092 . . . 4 (∀𝑦𝐴𝑧𝐵 ¬ 𝜓 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
63, 5bitri 275 . . 3 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
76notbii 320 . 2 (¬ ∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
8 dfrex2 3072 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ¬ ∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑)
9 dfrex2 3072 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
107, 8, 93bitr4i 303 1 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1539  wral 3060  wrex 3069  {csn 4625  cop 4631   ciun 4990   × cxp 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-iun 4992  df-opab 5205  df-xp 5690  df-rel 5691
This theorem is referenced by:  rexxp  5852  fsumvma  27258  cvmliftlem15  35304  filnetlem4  36383
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