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Theorem ceqsex 2893
 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 = A φ) ↔ 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 470 . . 3 ((x = A φ) → ψ)
41, 3exlimi 1803 . 2 (x(x = A φ) → ψ)
52biimprcd 216 . . . 4 (ψ → (x = Aφ))
61, 5alrimi 1765 . . 3 (ψx(x = Aφ))
7 ceqsex.2 . . . 4 A V
87isseti 2865 . . 3 x x = A
9 exintr 1614 . . 3 (x(x = Aφ) → (x x = Ax(x = A φ)))
106, 8, 9ee10 1376 . 2 (ψx(x = A φ))
114, 10impbii 180 1 (x(x = A φ) ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  ceqsexv  2894  ceqsex2  2895
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