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Theorem rexxp 5402
Description: Existential quantification restricted to a Cartesian product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
ralxp.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
rexxp (∃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 rexxp
StepHypRef Expression
1 iunxpconst 5314 . . 3 𝑦𝐴 ({𝑦} × 𝐵) = (𝐴 × 𝐵)
21rexeqi 3292 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝜑)
3 ralxp.1 . . 3 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
43rexiunxp 5400 . 2 (∃𝑥 𝑦𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
52, 4bitr3i 266 1 (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦𝐴𝑧𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wrex 3062  {csn 4317  cop 4323   ciun 4655   × cxp 5248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-iun 4657  df-opab 4848  df-xp 5256  df-rel 5257
This theorem is referenced by:  exopxfr  5403  fnrnov  6958  foov  6959  ovelimab  6963  el2xptp  7364  xpf1o  8282  xpwdomg  8650  hsmexlem2  9455  cnref1o  12030  vdwmc  15889  arwhoma  16902  txbas  21591  txkgen  21676  xrofsup  29873  elunirnmbfm  30655  madeval2  32273  rmxypairf1o  38000  unxpwdom3  38189
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