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Theorem rexsnsOLD 3401
 Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3400 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
rexsnsOLD (A 𝑉 → (x {A}φ[A / x]φ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem rexsnsOLD
StepHypRef Expression
1 sbc5 2781 . . 3 ([A / x]φx(x = A φ))
21a1i 9 . 2 (A 𝑉 → ([A / x]φx(x = A φ)))
3 df-rex 2306 . . 3 (x {A}φx(x {A} φ))
4 elsn 3382 . . . . 5 (x {A} ↔ x = A)
54anbi1i 431 . . . 4 ((x {A} φ) ↔ (x = A φ))
65exbii 1493 . . 3 (x(x {A} φ) ↔ x(x = A φ))
73, 6bitri 173 . 2 (x {A}φx(x = A φ))
82, 7syl6rbbr 188 1 (A 𝑉 → (x {A}φ[A / x]φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ 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:  rexsng  3403  r19.12sn  3427
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