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

Theorem exiftru 1889
Description: Rule of existential generalization, similar to universal generalization ax-gen 1720, but valid only if an individual exists. Its proof requires ax-6 1886 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1720, ax-4 1735 and this theorem alone, not requiring ax-7 1933 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
Ref Expression
exiftru.1 𝜑
Ref Expression
exiftru 𝑥𝜑

Proof of Theorem exiftru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1888 . 2 𝑥 𝑥 = 𝑦
2 exiftru.1 . . 3 𝜑
32a1i 11 . 2 (𝑥 = 𝑦𝜑)
41, 3eximii 1762 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-6 1886
This theorem depends on definitions:  df-bi 197  df-ex 1703
This theorem is referenced by:  19.2  1890  bj-extru  32629  ac6s6  33951
  Copyright terms: Public domain W3C validator