Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsexgv Structured version   Visualization version   GIF version

Theorem ceqsexgv 3647
 Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2141 and ax-12 2172. (Revised by Gino Giotto, 1-Dec-2023.)
Hypothesis
Ref Expression
ceqsexgv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsexgv (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsexgv
StepHypRef Expression
1 id 22 . 2 (𝑥 = 𝐴𝑥 = 𝐴)
2 ceqsexgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2cgsexg 3538 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893 This theorem is referenced by:  ceqsrexv  3649  clel3g  3654  elxp5  7622  xpsnen  8595  isssc  17084  metuel2  23169  isgrpo  28268  bj-finsumval0  34561  ismgmOLD  35122  brxrn  35620  pmapjat1  36983  dfatdmfcoafv2  43446
 Copyright terms: Public domain W3C validator