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Theorem nfcvf 2933
Description: If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. Usage of this theorem is discouraged because it depends on ax-13 2372. See nfcv 2904 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-ext 2704. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
Assertion
Ref Expression
nfcvf (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Proof of Theorem nfcvf
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1918 . 2 𝑤 ¬ ∀𝑥 𝑥 = 𝑦
2 nfv 1918 . . 3 𝑥 𝑤𝑧
3 elequ2 2122 . . 3 (𝑧 = 𝑦 → (𝑤𝑧𝑤𝑦))
42, 3dvelimnf 2453 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤𝑦)
51, 4nfcd 2892 1 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1540  wnfc 2884
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-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2372
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-nfc 2886
This theorem is referenced by:  nfcvf2  2934  nfrald  3369  ralcom2  3374  nfrmod  3429  nfreud  3430  nfrmo  3431  nfdisj  5127  nfcvb  5375  nfriotad  7377  nfixp  8911  axextnd  10586  axrepndlem2  10588  axrepnd  10589  axunndlem1  10590  axunnd  10591  axpowndlem2  10593  axpowndlem4  10595  axregndlem2  10598  axregnd  10599  axinfndlem1  10600  axinfnd  10601  axacndlem4  10605  axacndlem5  10606  axacnd  10607  axextdist  34771  bj-nfcsym  35779
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