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Theorem ralsns 3763
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsns (A V → (x {A}φ ↔ [̣A / xφ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem ralsns
StepHypRef Expression
1 sbc6g 3071 . 2 (A V → ([̣A / xφx(x = Aφ)))
2 df-ral 2619 . . 3 (x {A}φx(x {A} → φ))
3 elsn 3748 . . . . 5 (x {A} ↔ x = A)
43imbi1i 315 . . . 4 ((x {A} → φ) ↔ (x = Aφ))
54albii 1566 . . 3 (x(x {A} → φ) ↔ x(x = Aφ))
62, 5bitri 240 . 2 (x {A}φx(x = Aφ))
71, 6syl6rbbr 255 1 (A V → (x {A}φ ↔ [̣A / xφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  [̣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-ral 2619  df-v 2861  df-sbc 3047  df-sn 3741 This theorem is referenced by:  ralsng  3765  sbcsng  3783
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