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

Theorem 19.9d 2201
Description: A deduction version of one direction of 19.9 2203. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1792 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
Hypothesis
Ref Expression
19.9d.1 (𝜓 → Ⅎ𝑥𝜑)
Assertion
Ref Expression
19.9d (𝜓 → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9d
StepHypRef Expression
1 19.9d.1 . . 3 (𝜓 → Ⅎ𝑥𝜑)
21nfrd 1799 . 2 (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
3 sp 2180 . 2 (∀𝑥𝜑𝜑)
42, 3syl6 35 1 (𝜓 → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1541  wex 1787  wnf 1791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-ex 1788  df-nf 1792
This theorem is referenced by:  19.9t  2202  19.9ht  2319  spimt  2385  exdistrf  2446  equvel  2455  copsexgw  5378  copsexg  5379  oprabidw  7249  19.9d2rf  30544  copsex2d  35050  wl-exeq  35435  spd  46063
  Copyright terms: Public domain W3C validator