Theorem naecoms 2458

Description: A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)

Hypothesis

Ref	Expression
naecoms.1	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
naecoms	⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)

Proof of Theorem naecoms

Step	Hyp	Ref	Expression
1		aecom 2456	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
2		naecoms.1	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	sylnbir 334	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2213  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  sb9  2548  eujustALT  2597  nfcvf2  2949  axpowndlem2  10654  axsepg2  35733  axsepg4  35736  axnulg  35738  axpowg2  35740  axpowg3  35741  axtcond  37188  mh-setindnd  37247  wl-sbcom2d  38413  wl-mo2df  38422  wl-eudf  38424