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Theorem rexsng 3403
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 3401 . 2 (A 𝑉 → (x {A}φ[A / x]φ))
2 ralsng.1 . . 3 (x = A → (φψ))
32sbcieg 2789 . 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 1242   wcel 1390  wrex 2301  [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-bnd 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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-sn 3373
This theorem is referenced by:  rexsn  3406  rexprg  3413  rextpg  3415  iunxsng  3723  imasng  4633
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