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Theorem naecoms 2419
Description: A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
Hypothesis
Ref Expression
naecoms.1 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
Assertion
Ref Expression
naecoms (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Proof of Theorem naecoms
StepHypRef Expression
1 aecom 2417 . 2 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
2 naecoms.1 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2sylnbir 320 1 (¬ ∀𝑦 𝑦 = 𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-10 2132  ax-12 2160  ax-13 2355
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1818  df-nf 1823
This theorem is referenced by:  sb9  2527  eujustALT  2574  nfcvf2  2891  axpowndlem2  9533  wl-sbcom2d  33576  wl-mo2df  33584  wl-eudf  33586
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