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Theorem rexsns 3379
 Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
rexsns (x {A}φ[A / x]φ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rexsns
StepHypRef Expression
1 elsn 3361 . . . 4 (x {A} ↔ x = A)
21anbi1i 434 . . 3 ((x {A} φ) ↔ (x = A φ))
32exbii 1474 . 2 (x(x {A} φ) ↔ x(x = A φ))
4 df-rex 2286 . 2 (x {A}φx(x {A} φ))
5 sbc5 2760 . 2 ([A / x]φx(x = A φ))
63, 4, 53bitr4i 201 1 (x {A}φ[A / x]φ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281  [wsbc 2737  {csn 3346 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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-v 2533  df-sbc 2738  df-sn 3352 This theorem is referenced by: (None)
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