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

Theorem nfcvf2 2931
Description: If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. 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, 5-Dec-2016.) (New usage is discouraged.)
Assertion
Ref Expression
nfcvf2 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)

Proof of Theorem nfcvf2
StepHypRef Expression
1 nfcvf 2930 . 2 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
21naecoms 2432 1 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
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:  dfid3  5586  oprabid  7463  axrepndlem1  10630  axrepndlem2  10631  axrepnd  10632  axunnd  10634  axpowndlem3  10637  axpowndlem4  10638  axpownd  10639  axregndlem2  10641  axinfndlem1  10643  axinfnd  10644  axacndlem4  10648  axacndlem5  10649  axacnd  10650  bj-nfcsym  36882
  Copyright terms: Public domain W3C validator