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Theorem ceqsex 3545
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 477 . . 3 ((𝑥 = 𝐴𝜑) → 𝜓)
41, 3exlimi 2208 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓)
52biimprcd 251 . . . 4 (𝜓 → (𝑥 = 𝐴𝜑))
61, 5alrimi 2204 . . 3 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
7 ceqsex.2 . . . 4 𝐴 ∈ V
87isseti 3513 . . 3 𝑥 𝑥 = 𝐴
9 exintr 1886 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴𝜑)))
106, 8, 9mpisyl 21 . 2 (𝜓 → ∃𝑥(𝑥 = 𝐴𝜑))
114, 10impbii 210 1 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wex 1773  wnf 1777  wcel 2106  Vcvv 3499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-cleq 2817  df-clel 2897
This theorem is referenced by:  ceqsexv  3546  ceqsex2  3548
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