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Theorem rexsns 3400
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 3382 . . . 4 (x {A} ↔ x = A)
21anbi1i 431 . . 3 ((x {A} φ) ↔ (x = A φ))
32exbii 1493 . 2 (x(x {A} φ) ↔ x(x = A φ))
4 df-rex 2306 . 2 (x {A}φx(x {A} φ))
5 sbc5 2781 . 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 1242  wex 1378   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-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: (None)
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