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Theorem nfald 2322
Description: Deduction form of nfal 2317. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.)
Hypotheses
Ref Expression
nfald.1 𝑦𝜑
nfald.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfald (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfald
StepHypRef Expression
1 19.12 2321 . . 3 (∃𝑥𝑦𝜓 → ∀𝑦𝑥𝜓)
2 nfald.1 . . . 4 𝑦𝜑
3 nfald.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
43nfrd 1794 . . . 4 (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
52, 4alimd 2206 . . 3 (𝜑 → (∀𝑦𝑥𝜓 → ∀𝑦𝑥𝜓))
6 ax-11 2155 . . 3 (∀𝑦𝑥𝜓 → ∀𝑥𝑦𝜓)
71, 5, 6syl56 36 . 2 (𝜑 → (∃𝑥𝑦𝜓 → ∀𝑥𝑦𝜓))
87nfd 1793 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1782  wnf 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-or 847  df-ex 1783  df-nf 1787
This theorem is referenced by:  nfexd  2323  dvelimhw  2342  nfald2  2445  nfmodv  2554  nfeqd  2914  nfabdw  2927  nfraldw  3307  nfraldwOLD  3319  nfiotadw  6499  nfixpw  8910  axrepndlem1  10587  axrepndlem2  10588  axunnd  10591  axpowndlem2  10593  axpowndlem4  10595  axregndlem2  10598  axinfndlem1  10600  axinfnd  10601  axacndlem4  10605  axacndlem5  10606  axacnd  10607  bj-dvelimdv  35730  wl-mo2df  36435  wl-eudf  36437  wl-mo2t  36440  nfintd  47718
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