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Theorem sbcsng 3420
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
sbcsng (A 𝑉 → ([A / x]φx {A}φ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem sbcsng
StepHypRef Expression
1 ralsns 3399 . 2 (A 𝑉 → (x {A}φ[A / x]φ))
21bicomd 129 1 (A 𝑉 → ([A / x]φx {A}φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  ∀wral 2300  [wsbc 2758  {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-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-ral 2305  df-v 2553  df-sbc 2759  df-sn 3373 This theorem is referenced by: (None)
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