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Theorem bj-19.21t 32792
 Description: Proof of 19.21t 2071 from stdpc5t 32789. (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 32789 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
2 19.9t 2069 . . . 4 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
32imbi1d 331 . . 3 (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
4 19.38 1764 . . 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 1479  ∃wex 1702  Ⅎwnf 1706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045 This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708 This theorem is referenced by: (None)
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