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Theorem rexiunxp 5839
Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 5841, 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 317 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
32raliunxp 5838 . . . 4 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴𝑧𝐵 ¬ 𝜓)
4 ralnex 3070 . . . . 5 (∀𝑧𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧𝐵 𝜓)
54ralbii 3091 . . . 4 (∀𝑦𝐴𝑧𝐵 ¬ 𝜓 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
63, 5bitri 274 . . 3 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
76notbii 319 . 2 (¬ ∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
8 dfrex2 3071 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ¬ ∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑)
9 dfrex2 3071 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
107, 8, 93bitr4i 302 1 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1539  wral 3059  wrex 3068  {csn 4627  cop 4633   ciun 4996   × cxp 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-iun 4998  df-opab 5210  df-xp 5681  df-rel 5682
This theorem is referenced by:  rexxp  5841  fsumvma  26952  cvmliftlem15  34587  filnetlem4  35569
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