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Theorem ceqsexv2d 3216
 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 3215 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
54biimpri 217 . 2 (𝜓 → ∃𝑥(𝑥 = 𝐴𝜑))
6 exsimpr 1784 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥𝜑)
71, 5, 6mp2b 10 1 𝑥𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by:  2lgslem1  24919  griedg0prc  40488  1loopgrvd2  40718
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