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Theorem ralsng 3765
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
ralsng.1 (x = A → (φψ))
Assertion
Ref Expression
ralsng (A V → (x {A}φψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem ralsng
StepHypRef Expression
1 ralsns 3763 . 2 (A V → (x {A}φ ↔ [̣A / xφ))
2 ralsng.1 . . 3 (x = A → (φψ))
32sbcieg 3078 . 2 (A V → ([̣A / xφψ))
41, 3bitrd 244 1 (A V → (x {A}φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = 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-3an 936  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:  ralsn  3767  ralprg  3775  raltpg  3777  ralunsn  3879  iinxsng  4042
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