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Theorem nfnf 2196
Description: If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (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 1750 . 2 (Ⅎ𝑦𝜑 ↔ (∃𝑦𝜑 → ∀𝑦𝜑))
2 nfnf.1 . . . 4 𝑥𝜑
32nfex 2192 . . 3 𝑥𝑦𝜑
42nfal 2191 . . 3 𝑥𝑦𝜑
53, 4nfim 1865 . 2 𝑥(∃𝑦𝜑 → ∀𝑦𝜑)
61, 5nfxfr 1819 1 𝑥𝑦𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  nfnfc  2803  nfnfcALT  2804  bj-nfnfc  32978  bj-nfcf  33045
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