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Theorem r2ex 3196
Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020.)
Assertion
Ref Expression
r2ex (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem r2ex
StepHypRef Expression
1 r2al 3195 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
21r2exlem 3144 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wex 1782  wcel 2107  wrex 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-ral 3063  df-rex 3072
This theorem is referenced by:  reeanlem  3226  elxp2  5701  elinxp  6020  rnoprab2  7513  elrnmpores  7546  oeeu  8603  omxpenlem  9073  axcnre  11159  hash2prb  14433  hashle2prv  14439  pmtrrn2  19328  fsumvma  26716  umgredg  28398  fusgr2wsp2nb  29587  spanuni  30797  5oalem7  30913  3oalem3  30917  trsp2cyc  32282  fmla0xp  34374  elfuns  34887  ellines  35124  dalem20  38564  diblsmopel  40042  iunrelexpuztr  42470  sprssspr  46149  prprelb  46184
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