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Theorem rexdifsn 4793
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 4785 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 624 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
3 anass 468 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
42, 3bitri 275 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
54rexbii2 3089 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2107  wne 2939  wrex 3069  cdif 3947  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rex 3070  df-v 3481  df-dif 3953  df-sn 4626
This theorem is referenced by:  symgfix2  19435  usgr2pth0  29786  wspniunwspnon  29944  dihatexv  41341  lcfl8b  41507
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