MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.9 Structured version   Visualization version   GIF version

Theorem 19.9 2199
Description: A wff may be existentially quantified with a variable not free in it. Version of 19.3 2196 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1988 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
Hypothesis
Ref Expression
19.9.1 𝑥𝜑
Assertion
Ref Expression
19.9 (∃𝑥𝜑𝜑)

Proof of Theorem 19.9
StepHypRef Expression
1 19.9.1 . 2 𝑥𝜑
2 19.9t 2198 . 2 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2ax-mp 5 1 (∃𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1782  wnf 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787
This theorem is referenced by:  exlimd  2212  19.19  2223  19.36  2224  19.41  2229  19.44  2231  19.45  2232  19.9h  2283  eeor  2330  dfid3  5578  bnj1189  34020  bj-exexbiex  35578  bj-exalbial  35580  ax6e2ndeq  43320  e2ebind  43324  ax6e2ndeqVD  43670  e2ebindVD  43673  e2ebindALT  43690  ax6e2ndeqALT  43692
  Copyright terms: Public domain W3C validator