New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  rexsns GIF version

Theorem rexsns 3764
 Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
rexsns (A V → (x {A}φ ↔ [̣A / xφ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem rexsns
StepHypRef Expression
1 sbc5 3070 . . 3 ([̣A / xφx(x = A φ))
21a1i 10 . 2 (A V → ([̣A / xφx(x = A φ)))
3 df-rex 2620 . . 3 (x {A}φx(x {A} φ))
4 elsn 3748 . . . . 5 (x {A} ↔ x = A)
54anbi1i 676 . . . 4 ((x {A} φ) ↔ (x = A φ))
65exbii 1582 . . 3 (x(x {A} φ) ↔ x(x = A φ))
73, 6bitri 240 . 2 (x {A}φx(x = A φ))
82, 7syl6rbbr 255 1 (A V → (x {A}φ ↔ [̣A / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  [̣wsbc 3046  {csn 3737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-sn 3741 This theorem is referenced by:  rexsng  3766  r19.12sn  3789
 Copyright terms: Public domain W3C validator