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Theorem euexALTOLD 2643
 Description: Obsolete proof of euex 2596 as of 31-Dec-2022. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
euexALTOLD (∃!𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem euexALTOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1873 . . 3 𝑦𝜑
21eu1 2641 . 2 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
3 exsimpl 1831 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) → ∃𝑥𝜑)
42, 3sylbi 209 1 (∃!𝑥𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387  ∀wal 1505  ∃wex 1742  [wsb 2015  ∃!weu 2583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-11 2093  ax-12 2106 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584 This theorem is referenced by: (None)
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