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Theorem nfrd 1792
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 1785 . 2 (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
31, 2sylib 220 1 (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wex 1780  wnf 1784
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 209  df-nf 1785
This theorem is referenced by:  19.38a  1840  19.38b  1841  nfimd  1895  nf5r  2193  19.9d  2203  nfald  2347  exists2  2746  eusv2i  5271  bj-nfimt  33979  bj-nfald  34446
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