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Theorem ceqsexv2d 3496
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
Hypotheses
Ref Expression
ceqsexv2d.1 𝐴 ∈ V
ceqsexv2d.2 (𝑥 = 𝐴 → (𝜑𝜓))
ceqsexv2d.3 𝜓
Assertion
Ref Expression
ceqsexv2d 𝑥𝜑
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexv2d
StepHypRef Expression
1 ceqsexv2d.3 . 2 𝜓
2 ceqsexv2d.1 . . . 4 𝐴 ∈ V
3 ceqsexv2d.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3ceqsexv 3493 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
54biimpri 227 . 2 (𝜓 → ∃𝑥(𝑥 = 𝐴𝜑))
6 exsimpr 1873 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥𝜑)
71, 5, 6mp2b 10 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2816
This theorem is referenced by:  en0  8891  en0OLD  8892  en0r  8894  ensn1  8895  ensn1OLD  8896  0domg  8978  tz9.1  9599  cplem2  9760  karden  9765  pwmnd  18682  2lgslem1  26664  griedg0prc  27998  1loopgrvd2  28237  bnj150  33249  fnchoice  42967  nfermltl8rev  45652  nfermltl2rev  45653  nfermltlrev  45654  nn0mnd  45831  rrx2xpreen  46523
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