MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfcvf2 Structured version   Visualization version   GIF version

Theorem nfcvf2 2938
Description: If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
nfcvf2 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)

Proof of Theorem nfcvf2
StepHypRef Expression
1 nfcvf 2937 . 2 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
21naecoms 2465 1 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629  wnfc 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-cleq 2764  df-clel 2767  df-nfc 2902
This theorem is referenced by:  dfid3  5158  oprabid  6822  axrepndlem1  9616  axrepndlem2  9617  axrepnd  9618  axunnd  9620  axpowndlem3  9623  axpowndlem4  9624  axpownd  9625  axregndlem2  9627  axinfndlem1  9629  axinfnd  9630  axacndlem4  9634  axacndlem5  9635  axacnd  9636  bj-nfcsym  33215
  Copyright terms: Public domain W3C validator