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Theorem nfcvf 2930
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 2375. See nfcv 2903 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 2706. (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 1912 . 2 𝑤 ¬ ∀𝑥 𝑥 = 𝑦
2 nfv 1912 . . 3 𝑥 𝑤𝑧
3 elequ2 2121 . . 3 (𝑧 = 𝑦 → (𝑤𝑧𝑤𝑦))
42, 3dvelimnf 2456 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤𝑦)
51, 4nfcd 2896 1 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-nfc 2890
This theorem is referenced by:  nfcvf2  2931  nfrald  3370  ralcom2  3375  nfrmod  3429  nfreud  3430  nfrmo  3431  nfdisj  5128  nfcvb  5382  nfriotad  7399  nfixp  8956  axextnd  10629  axrepndlem2  10631  axrepnd  10632  axunndlem1  10633  axunnd  10634  axpowndlem2  10636  axpowndlem4  10638  axregndlem2  10641  axregnd  10642  axinfndlem1  10643  axinfnd  10644  axacndlem4  10648  axacndlem5  10649  axacnd  10650  axsepg2  35075  axsepg2ALT  35076  axnulg  35085  axextdist  35781  bj-nfcsym  36882
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