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Theorem nfcvf 2199
 Description: If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
nfcvf (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)

Proof of Theorem nfcvf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2178 . 2 𝑥𝑧
2 nfcv 2178 . 2 𝑧𝑦
3 id 19 . 2 (𝑧 = 𝑦𝑧 = 𝑦)
41, 2, 3dvelimc 2198 1 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241  Ⅎwnfc 2165 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167 This theorem is referenced by:  nfcvf2  2200
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