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Theorem nfexd 2322
Description: If 𝑥 is not free in 𝜓, then it is not free in 𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
nfald.1 𝑦𝜑
nfald.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfexd (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfexd
StepHypRef Expression
1 df-ex 1782 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2 nfald.1 . . . 4 𝑦𝜑
3 nfald.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
43nfnd 1861 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
52, 4nfald 2321 . . 3 (𝜑 → Ⅎ𝑥𝑦 ¬ 𝜓)
65nfnd 1861 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
71, 6nfxfrd 1856 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1539  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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1782  df-nf 1786
This theorem is referenced by:  nfmod2  2552  nfmodv  2553  nfeudw  2585  nfeld  2914  nfopabd  5216  nfttrcld  9704  axrepndlem1  10586  axrepndlem2  10587  axunndlem1  10589  axunnd  10590  axpowndlem2  10592  axpowndlem3  10593  axpowndlem4  10594  axregndlem2  10597  axinfndlem1  10599  axinfnd  10600  axacndlem4  10604  axacndlem5  10605  axacnd  10606  19.9d2rf  31706  hbexg  43307
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