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Theorem nfnf 2144
 Description: If 𝑥 is not free in 𝜑, it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
Hypothesis
Ref Expression
nfnf.1 𝑥𝜑
Assertion
Ref Expression
nfnf 𝑥𝑦𝜑

Proof of Theorem nfnf
StepHypRef Expression
1 df-nf 1701 . 2 (Ⅎ𝑦𝜑 ↔ (∃𝑦𝜑 → ∀𝑦𝜑))
2 nfnf.1 . . . 4 𝑥𝜑
32nfex 2140 . . 3 𝑥𝑦𝜑
42nfal 2139 . . 3 𝑥𝑦𝜑
53, 4nfim 1813 . 2 𝑥(∃𝑦𝜑 → ∀𝑦𝜑)
61, 5nfxfr 1771 1 𝑥𝑦𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  nfnfc  2760  nfnfcALT  2761  bj-nfnfc  32047  bj-nfcf  32112
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