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Theorem pm10.52 39090
Description: Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.52 (∃𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ 𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm10.52
StepHypRef Expression
1 19.23v 2023 . 2 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
2 pm5.5 350 . 2 (∃𝑥𝜑 → ((∃𝑥𝜑𝜓) ↔ 𝜓))
31, 2syl5bb 272 1 (∃𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by: (None)
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