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Theorem 19.21tOLD 2249
Description: Obsolete proof of 19.21t 2111 as of 6-Oct-2021. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
19.21tOLD (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21tOLD
StepHypRef Expression
1 19.21t-1OLD 2248 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
2 19.9tOLD 2240 . . . 4 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 330 . . 3 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
4 19.38 1806 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
53, 4syl6bir 244 . 2 (Ⅎ𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓)))
61, 5impbid 202 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wex 1744  wnfOLD 1749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nfOLD 1761
This theorem is referenced by:  19.21OLD  2250  19.23tOLD  2254  nfimdOLD  2262
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