Theorem naecoms 2459

Description: A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2402. (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 2457	. 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)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-10 2174  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-nf 1812

This theorem is referenced by:  sb9  2549  eujustALT  2598  nfcvf2  2950  axpowndlem2  10582  axsepg2  35519  axsepg4  35522  axnulg  35524  axpowg2  35526  axpowg3  35527  axtcond  36955  mh-setindnd  37014  wl-sbcom2d  38182  wl-mo2df  38191  wl-eudf  38193