Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexiunxp Structured version   Visualization version   GIF version

Theorem rexiunxp 5295
 Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 5297, 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 307 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
32raliunxp 5294 . . . 4 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴𝑧𝐵 ¬ 𝜓)
4 ralnex 3021 . . . . 5 (∀𝑧𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧𝐵 𝜓)
54ralbii 3009 . . . 4 (∀𝑦𝐴𝑧𝐵 ¬ 𝜓 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
63, 5bitri 264 . . 3 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
76notbii 309 . 2 (¬ ∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
8 dfrex2 3025 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ¬ ∀𝑥 𝑦𝐴 ({𝑦} × 𝐵) ¬ 𝜑)
9 dfrex2 3025 . 2 (∃𝑦𝐴𝑧𝐵 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ ∃𝑧𝐵 𝜓)
107, 8, 93bitr4i 292 1 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1523  ∀wral 2941  ∃wrex 2942  {csn 4210  ⟨cop 4216  ∪ ciun 4552   × cxp 5141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-iun 4554  df-opab 4746  df-xp 5149  df-rel 5150 This theorem is referenced by:  rexxp  5297  fsumvma  24983  cvmliftlem15  31406  filnetlem4  32501
 Copyright terms: Public domain W3C validator