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Theorem 19.38aOLD 1936
 Description: Obsolete version of 19.38a 1935 as of 9-Jul-2022. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.38aOLD (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Proof of Theorem 19.38aOLD
StepHypRef Expression
1 19.38 1934 . 2 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
2 df-nf 1880 . . 3 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
3 alim 1906 . . . 4 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
4 imim1 83 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝜑) → ((∀𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
53, 4syl5 34 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
62, 5sylbi 209 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
71, 6impbid2 218 1 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651  ∃wex 1875  Ⅎwnf 1879 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905 This theorem depends on definitions:  df-bi 199  df-ex 1876  df-nf 1880 This theorem is referenced by: (None)
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