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

Theorem naecoms 2443
 Description: A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
Hypothesis
Ref Expression
naecoms.1 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
Assertion
Ref Expression
naecoms (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Proof of Theorem naecoms
StepHypRef Expression
1 aecom 2441 . 2 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
2 naecoms.1 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2sylnbir 334 1 (¬ ∀𝑦 𝑦 = 𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by:  sb9  2541  eujustALT  2635  nfcvf2  2985  axpowndlem2  10013  wl-sbcom2d  34955  wl-mo2df  34964  wl-eudf  34966
 Copyright terms: Public domain W3C validator