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Theorem ralxp 5172
Description: Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
ralxp.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
ralxp (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem ralxp
StepHypRef Expression
1 iunxpconst 5087 . . 3 𝑦𝐴 ({𝑦} × 𝐵) = (𝐴 × 𝐵)
21raleqi 3118 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥 ∈ (𝐴 × 𝐵)𝜑)
3 ralxp.1 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
43raliunxp 5170 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
52, 4bitr3i 264 1 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wral 2895  {csn 4124  cop 4130   ciun 4449   × cxp 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-iun 4451  df-opab 4638  df-xp 5033  df-rel 5034
This theorem is referenced by:  ralxpf  5177  issref  5414  ffnov  6639  eqfnov  6641  funimassov  6686  f1stres  7058  f2ndres  7059  ecopover  7715  ecopoverOLD  7716  xpf1o  7984  xpwdomg  8350  rankxplim  8602  imasaddfnlem  15959  imasvscafn  15968  comfeq  16137  isssc  16251  isfuncd  16296  cofucl  16319  funcres2b  16328  evlfcl  16633  uncfcurf  16650  yonedalem3  16691  yonedainv  16692  efgval2  17908  srgfcl  18286  txbas  21127  hausdiag  21205  tx1stc  21210  txkgen  21212  xkococn  21220  cnmpt21  21231  xkoinjcn  21247  tmdcn2  21650  clssubg  21669  qustgplem  21681  txmetcnp  22109  txmetcn  22110  qtopbaslem  22319  bndth  22512  cxpcn3  24233  dvdsmulf1o  24664  fsumdvdsmul  24665  xrofsup  28716  txpcon  30261  cvmlift2lem1  30331  cvmlift2lem12  30343  mclsax  30513  f1opr  32472  ismtyhmeolem  32556  dih1dimatlem  35419  ffnaov  39712  ovn0ssdmfun  41538  plusfreseq  41543
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