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Theorem 19.21t 1569
Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
Assertion
Ref Expression
19.21t (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21t
StepHypRef Expression
1 df-nf 1448 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 19.21ht 1568 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
31, 2sylbi 120 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1340  wnf 1447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-4 1497  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-nf 1448
This theorem is referenced by:  19.21  1570  nfimd  1572  equs5or  1817  sbal1yz  1988  r19.21t  2539  ceqsalt  2750  sbciegft  2979
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