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Theorem ralxp 5676
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 5588 . . 3 𝑦𝐴 ({𝑦} × 𝐵) = (𝐴 × 𝐵)
21raleqi 3362 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑥 ∈ (𝐴 × 𝐵)𝜑)
3 ralxp.1 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
43raliunxp 5674 . 2 (∀𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
52, 4bitr3i 280 1 (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wral 3106  {csn 4525  cop 4531   ciun 4881   × cxp 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4883  df-opab 5093  df-xp 5525  df-rel 5526
This theorem is referenced by:  ralxpf  5681  reu3op  6111  f1opr  7189  ffnov  7257  eqfnov  7259  funimassov  7305  f1stres  7695  f2ndres  7696  ecopover  8384  xpf1o  8663  xpwdomg  9033  rankxplim  9292  imasaddfnlem  16793  imasvscafn  16802  comfeq  16968  isssc  17082  isfuncd  17127  cofucl  17150  funcres2b  17159  evlfcl  17464  uncfcurf  17481  yonedalem3  17522  yonedainv  17523  efgval2  18842  srgfcl  19258  txbas  22172  hausdiag  22250  tx1stc  22255  txkgen  22257  xkococn  22265  cnmpt21  22276  xkoinjcn  22292  tmdcn2  22694  clssubg  22714  qustgplem  22726  txmetcnp  23154  txmetcn  23155  qtopbaslem  23364  bndth  23563  cxpcn3  25337  dvdsmulf1o  25779  fsumdvdsmul  25780  xrofsup  30518  txpconn  32589  cvmlift2lem1  32659  cvmlift2lem12  32671  mclsax  32926  ismtyhmeolem  35239  dih1dimatlem  38622  ffnaov  43750  ovn0ssdmfun  44382  plusfreseq  44387
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