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Theorem naecoms 2443
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 2441 . 2 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
2 naecoms.1 . 2 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2sylnbir 332 1 (¬ ∀𝑦 𝑦 = 𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776
This theorem is referenced by:  sb9  2554  eujustALT  2650  nfcvf2  3005  axpowndlem2  10008  wl-sbcom2d  34678  wl-mo2df  34687  wl-eudf  34689
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