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Theorem ceqsex 3529
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Jan-2025.)
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 alinexa 1842 . . 3 (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝐴𝜑))
2 ceqsex.1 . . . . 5 𝑥𝜓
32nfn 1856 . . . 4 𝑥 ¬ 𝜓
4 ceqsex.2 . . . 4 𝐴 ∈ V
5 ceqsex.3 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
65notbid 318 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
73, 4, 6ceqsal 3518 . . 3 (∀𝑥(𝑥 = 𝐴 → ¬ 𝜑) ↔ ¬ 𝜓)
81, 7bitr3i 277 . 2 (¬ ∃𝑥(𝑥 = 𝐴𝜑) ↔ ¬ 𝜓)
98con4bii 321 1 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wex 1778  wnf 1782  wcel 2107  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-clel 2815
This theorem is referenced by:  ceqsex2  3534
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