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Theorem rexsng 3386
 Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
Hypothesis
Ref Expression
ralsng.1 (x = A → (φψ))
Assertion
Ref Expression
rexsng (A 𝑉 → (x {A}φψ))
Distinct variable groups:   x,A   ψ,x
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem rexsng
StepHypRef Expression
1 rexsnsOLD 3384 . 2 (A 𝑉 → (x {A}φ[A / x]φ))
2 ralsng.1 . . 3 (x = A → (φψ))
32sbcieg 2772 . 2 (A 𝑉 → ([A / x]φψ))
41, 3bitrd 177 1 (A 𝑉 → (x {A}φψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∃wrex 2285  [wsbc 2741  {csn 3350 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-sbc 2742  df-sn 3356 This theorem is referenced by:  rexsn  3389  rexprg  3396  rextpg  3398  iunxsng  3706  imasng  4617
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