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Theorem nfrd 1887
Description: Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)
Hypothesis
Ref Expression
nfrd.1 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrd (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))

Proof of Theorem nfrd
StepHypRef Expression
1 nfrd.1 . 2 (𝜑 → Ⅎ𝑥𝜓)
2 df-nf 1880 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
31, 2sylib 210 1 (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651  wex 1875  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-nf 1880
This theorem is referenced by:  19.38a  1935  19.38b  1937  nfimd  1993  19.9d  2236  nfald  2351  eusv2i  5064  bj-exlalrim  33109  bj-nfimt  33122
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