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Theorem nfnf 2346
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 1786 . 2 (Ⅎ𝑦𝜑 ↔ (∃𝑦𝜑 → ∀𝑦𝜑))
2 nfnf.1 . . . 4 𝑥𝜑
32nfex 2344 . . 3 𝑥𝑦𝜑
42nfal 2343 . . 3 𝑥𝑦𝜑
53, 4nfim 1897 . 2 𝑥(∃𝑦𝜑 → ∀𝑦𝜑)
61, 5nfxfr 1854 1 𝑥𝑦𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  nfnfc  2991  bj-nfcf  34327
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