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Theorem rexdifsn 4798
Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
Assertion
Ref Expression
rexdifsn (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))

Proof of Theorem rexdifsn
StepHypRef Expression
1 eldifsn 4791 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 625 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
3 anass 470 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
42, 3bitri 275 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
54rexbii2 3091 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  wne 2941  wrex 3071  cdif 3946  {csn 4629
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  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rex 3072  df-v 3477  df-dif 3952  df-sn 4630
This theorem is referenced by:  symgfix2  19284  usgr2pth0  29022  wspniunwspnon  29177  dihatexv  40209  lcfl8b  40375
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