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Theorem 19.9d 2203
 Description: A deduction version of one direction of 19.9 2205. (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 1785 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 1792 . 2 (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
3 sp 2182 . 2 (∀𝑥𝜑𝜑)
42, 3syl6 35 1 (𝜓 → (∃𝑥𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177 This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785 This theorem is referenced by:  19.9t  2204  19.9ht  2339  spimt  2404  exdistrf  2469  equvel  2479  copsexgw  5357  copsexg  5358  oprabidw  7164  19.9d2rf  30220  copsex2d  34448  wl-exeq  34815  spd  44968
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