Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  euexALT Structured version   Visualization version   GIF version

Theorem euexALT 2540
 Description: Alternate proof of euex 2522. Shorter but uses more axioms. (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
euexALT (∃!𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem euexALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1883 . . 3 𝑦𝜑
21eu1 2539 . 2 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
3 exsimpl 1835 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) → ∃𝑥𝜑)
42, 3sylbi 207 1 (∃!𝑥𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1521  ∃wex 1744  [wsb 1937  ∃!weu 2498 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator