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Theorem 19.9dOLDOLD 2245
 Description: Obsolete version of 19.9d 2244 as of 8-Jul-2022. (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 1883 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.9d.1 (𝜓 → Ⅎ𝑥𝜑)
Assertion
Ref Expression
19.9dOLDOLD (𝜓 → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9dOLDOLD
StepHypRef Expression
1 19.9d.1 . . 3 (𝜓 → Ⅎ𝑥𝜑)
2 df-nf 1883 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
31, 2sylib 210 . 2 (𝜓 → (∃𝑥𝜑 → ∀𝑥𝜑))
4 sp 2224 . 2 (∀𝑥𝜑𝜑)
53, 4syl6 35 1 (𝜓 → (∃𝑥𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1654  ∃wex 1878  Ⅎwnf 1882 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220 This theorem depends on definitions:  df-bi 199  df-ex 1879  df-nf 1883 This theorem is referenced by: (None)
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