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

Theorem eximd 2123
Description: Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1801. (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 2103 . 2 (𝜑 → ∀𝑥𝜑)
3 eximd.2 . 2 (𝜑 → (𝜓𝜒))
42, 3eximdh 1831 1 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1744  wnf 1748
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-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750
This theorem is referenced by:  exlimd  2125  19.41  2141  19.42-1  2142  2ax6elem  2477  mopick2  2569  2euex  2573  reximd2a  3042  ssrexf  3698  axpowndlem3  9459  axregndlem1  9462  axregnd  9464  spc2ed  29440  padct  29625  finminlem  32437  bj-mo3OLD  32957  wl-euequ1f  33486  pmapglb2xN  35376  disjinfi  39694  infrpge  39880  fsumiunss  40125  islpcn  40189  stoweidlem27  40562  stoweidlem34  40569  stoweidlem35  40570  sge0rpcpnf  40956
  Copyright terms: Public domain W3C validator