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Theorem rexdifsn 3843
 Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
Assertion
Ref Expression
rexdifsn (x (A {B})φx A (xB φ))

Proof of Theorem rexdifsn
StepHypRef Expression
1 eldifsn 3839 . . . 4 (x (A {B}) ↔ (x A xB))
21anbi1i 676 . . 3 ((x (A {B}) φ) ↔ ((x A xB) φ))
3 anass 630 . . 3 (((x A xB) φ) ↔ (x A (xB φ)))
42, 3bitri 240 . 2 ((x (A {B}) φ) ↔ (x A (xB φ)))
54rexbii2 2643 1 (x (A {B})φx A (xB φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ∖ cdif 3206  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-sn 3741 This theorem is referenced by: (None)
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