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

Theorem nfcvf2 2934
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 2372. See nfcv 2904 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 2933 . 2 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
21naecoms 2429 1 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
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:  dfid3  5578  oprabid  7441  axrepndlem1  10587  axrepndlem2  10588  axrepnd  10589  axunnd  10591  axpowndlem3  10594  axpowndlem4  10595  axpownd  10596  axregndlem2  10598  axinfndlem1  10600  axinfnd  10601  axacndlem4  10605  axacndlem5  10606  axacnd  10607  bj-nfcsym  35779
  Copyright terms: Public domain W3C validator