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Theorem 19.21t-1OLD 2374
Description: One direction of the bi-conditional in 19.21t 2229. Unlike the reverse implication, it does not depend on ax-10 2174. Obsolete as of 6-Oct-2021 (Contributed by Wolf Lammen, 4-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.21t-1OLD (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))

Proof of Theorem 19.21t-1OLD
StepHypRef Expression
1 nfrOLD 2350 . 2 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
2 alim 1886 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
31, 2syl9 77 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wnfOLD 1857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nfOLD 1869
This theorem is referenced by:  19.21tOLD  2375  stdpc5OLDOLD  2379
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