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Theorem 19.9d 2058
Description: A deduction version of one direction of 19.9 2060. (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 1701 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
31, 2sylib 207 . 2 (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
4 sp 2041 . 2 (∀𝑥𝜑𝜑)
53, 4syl6 34 1 (𝜓 → (∃𝑥𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wex 1695  wnf 1699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701
This theorem is referenced by:  19.9t  2059  exdistrf  2321  equvel  2335  copsexg  4882  19.9d2rf  28702  wl-exeq  32500  spd  42223
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