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Theorem rexdifsn 3490
 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 3486 . . . 4 (x (A ∖ {B}) ↔ (x A xB))
21anbi1i 431 . . 3 ((x (A ∖ {B}) φ) ↔ ((x A xB) φ))
3 anass 381 . . 3 (((x A xB) φ) ↔ (x A (xB φ)))
42, 3bitri 173 . 2 ((x (A ∖ {B}) φ) ↔ (x A (xB φ)))
54rexbii2 2329 1 (x (A ∖ {B})φx A (xB φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1390   ≠ wne 2201  ∃wrex 2301   ∖ cdif 2908  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-rex 2306  df-v 2553  df-dif 2914  df-sn 3373 This theorem is referenced by: (None)
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