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Theorem naecoms 2432
Description: A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 2430 . 2 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
2 naecoms.1 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2sylnbir 331 1 (¬ ∀𝑦 𝑦 = 𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535
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-10 2139  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781
This theorem is referenced by:  sb9  2522  eujustALT  2570  nfcvf2  2931  axpowndlem2  10636  axnulg  35085  wl-sbcom2d  37542  wl-mo2df  37551  wl-eudf  37553
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