Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ceqsexv GIF version

Theorem ceqsexv 2729
 Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
Hypotheses
Ref Expression
ceqsexv.1 𝐴 ∈ V
ceqsexv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsexv (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexv
StepHypRef Expression
1 nfv 1509 . 2 𝑥𝜓
2 ceqsexv.1 . 2 𝐴 ∈ V
3 ceqsexv.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3ceqsex 2728 1 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2690 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2692 This theorem is referenced by:  ceqsex3v  2732  gencbvex  2736  sbhypf  2739  euxfr2dc  2874  inuni  4089  eqvinop  4174  onm  4332  uniuni  4381  opeliunxp  4604  elvvv  4612  rexiunxp  4691  imai  4905  coi1  5064  abrexco  5670  opabex3d  6029  opabex3  6030  mapsnen  6715  xpsnen  6725  xpcomco  6730  xpassen  6734
 Copyright terms: Public domain W3C validator