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Theorem rexsn 4598
Description: Convert an existential quantification restricted to a singleton to a substitution. (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
ralsn.1 𝐴 ∈ V
ralsn.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsn (∃𝑥 ∈ {𝐴}𝜑𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexsn
StepHypRef Expression
1 ralsn.1 . 2 𝐴 ∈ V
2 ralsn.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32rexsng 4590 . 2 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑𝜓))
41, 3ax-mp 5 1 (∃𝑥 ∈ {𝐴}𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wrex 3062  Vcvv 3408  {csn 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-v 3410  df-sn 4542
This theorem is referenced by:  elsnres  5891  oarec  8290  snec  8462  zornn0g  10119  fpwwe2lem12  10256  elreal  10745  hashge2el2difr  14047  vdwlem6  16539  pmatcollpw3fi1  21685  restsn  22067  snclseqg  23013  ust0  23117  grplsm0l  31305  esum2dlem  31772  eulerpartlemgh  32057  eldm3  33447  0slt1s  33760  made0  33794  cofcutr  33829  poimirlem28  35542  heiborlem3  35708
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