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 2056
Description: A deduction version of one direction of 19.9 2058. (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 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
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)
2 df-nf 1700 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
31, 2sylib 206 . 2 (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
4 sp 2039 . 2 (∀𝑥𝜑𝜑)
53, 4syl6 34 1 (𝜓 → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700
This theorem is referenced by:  19.9t  2057  exdistrf  2317  equvel  2331  copsexg  4873  19.9d2rf  28505  wl-exeq  32300
  Copyright terms: Public domain W3C validator