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

Theorem eximd 2249
Description: Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1928. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
eximd.1 𝑥𝜑
eximd.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
eximd (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Proof of Theorem eximd
StepHypRef Expression
1 eximd.1 . . 3 𝑥𝜑
21nf5ri 2227 . 2 (𝜑 → ∀𝑥𝜑)
3 eximd.2 . 2 (𝜑 → (𝜓𝜒))
42, 3eximdh 1961 1 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1874  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-ex 1875  df-nf 1879
This theorem is referenced by:  exlimd  2251  19.41  2268  19.42-1OLD  2269  sbimd  2275  2ax6elem  2541  mopick2  2662  2euex  2666  reximd2a  3159  ssrexf  3827  axpowndlem3  9678  axregndlem1  9681  axregnd  9683  spc2ed  29789  padct  29967  finminlem  32777  bj-mo3OLD  33282  wl-euequf  33802  pmapglb2xN  35749  disjinfi  40051  infrpge  40229  fsumiunss  40469  islpcn  40533  stoweidlem34  40912  stoweidlem35  40913  sge0rpcpnf  41299
  Copyright terms: Public domain W3C validator