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Theorem rexdifsn 4556
 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 4550 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21anbi1i 617 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝜑))
3 anass 462 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
42, 3bitri 267 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵}) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝜑)))
54rexbii2 3222 1 (∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∈ wcel 2107   ≠ wne 2969  ∃wrex 3091   ∖ cdif 3789  {csn 4398 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-rex 3096  df-v 3400  df-dif 3795  df-sn 4399 This theorem is referenced by:  symgfix2  18219  usgr2pth0  27117  wspniunwspnon  27303  dihatexv  37494  lcfl8b  37660
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