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

Theorem exiftru 2060
 Description: Rule of existential generalization, similar to universal generalization ax-gen 1870, but valid only if an individual exists. Its proof requires ax-6 2057 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1870, ax-4 1885 and this theorem alone, not requiring ax-7 2093 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
Hypothesis
Ref Expression
exiftru.1 𝜑
Assertion
Ref Expression
exiftru 𝑥𝜑

Proof of Theorem exiftru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 2059 . 2 𝑥 𝑥 = 𝑦
2 exiftru.1 . . 3 𝜑
32a1i 11 . 2 (𝑥 = 𝑦𝜑)
41, 3eximii 1912 1 𝑥𝜑
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-6 2057 This theorem depends on definitions:  df-bi 197  df-ex 1853 This theorem is referenced by:  19.2  2061  bj-extru  32991  ac6s6  34312
 Copyright terms: Public domain W3C validator