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Theorem ceqsex 2698
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
ceqsex.1 𝑥𝜓
ceqsex.2 𝐴 ∈ V
ceqsex.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsex (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ceqsex
StepHypRef Expression
1 ceqsex.1 . . 3 𝑥𝜓
2 ceqsex.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
32biimpa 294 . . 3 ((𝑥 = 𝐴𝜑) → 𝜓)
41, 3exlimi 1558 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓)
52biimprcd 159 . . . 4 (𝜓 → (𝑥 = 𝐴𝜑))
61, 5alrimi 1487 . . 3 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
7 ceqsex.2 . . . 4 𝐴 ∈ V
87isseti 2668 . . 3 𝑥 𝑥 = 𝐴
9 exintr 1598 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴𝜑)))
106, 8, 9mpisyl 1407 . 2 (𝜓 → ∃𝑥(𝑥 = 𝐴𝜑))
114, 10impbii 125 1 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1314   = wceq 1316  wnf 1421  wex 1453  wcel 1465  Vcvv 2660
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  ceqsexv  2699  ceqsex2  2700
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