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

Theorem aleximi 1907
Description: A variant of al2imi 1891: instead of applying 𝑥 quantifiers to the final implication, replace them with 𝑥. A shorter proof is possible using nfa1 2184, sps 2209 and eximd 2241, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
Hypothesis
Ref Expression
aleximi.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
aleximi (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Proof of Theorem aleximi
StepHypRef Expression
1 aleximi.1 . . . . 5 (𝜑 → (𝜓𝜒))
21con3d 149 . . . 4 (𝜑 → (¬ 𝜒 → ¬ 𝜓))
32al2imi 1891 . . 3 (∀𝑥𝜑 → (∀𝑥 ¬ 𝜒 → ∀𝑥 ¬ 𝜓))
4 alnex 1854 . . 3 (∀𝑥 ¬ 𝜒 ↔ ¬ ∃𝑥𝜒)
5 alnex 1854 . . 3 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
63, 4, 53imtr3g 284 . 2 (∀𝑥𝜑 → (¬ ∃𝑥𝜒 → ¬ ∃𝑥𝜓))
76con4d 115 1 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629  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
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  alexbii  1908  exim  1909  exanOLD  1940  eximdh  1942  19.29  1953  19.29r  1954  19.35  1957  19.25  1960  19.30  1961  19.40b  1967  speimfw  2045  aeveq  2139  nfeqf2  2452  2ax6elem  2597  mo3  2656  mopick  2684  2mo  2700  ssopab2  5134  ssoprab2  6858  axextnd  9615  bj-2exim  32932  bj-exalimi  32949  bj-sb56  32976  wl-mo3t  33692  pm10.56  39095  2exim  39104
  Copyright terms: Public domain W3C validator