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Theorem bj-19.21t 31814
Description: Proof of 19.21t 2058 from stdpc5t 31811. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-19.21t (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-19.21t
StepHypRef Expression
1 stdpc5t 31811 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
2 19.9t 2056 . . . 4 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 329 . . 3 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
4 19.38 1755 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
53, 4syl6bir 242 . 2 (Ⅎ𝑥𝜑 → ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓)))
61, 5impbid 200 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
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