Theorem naecoms 2461

Description: A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2404. (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 2459	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
2		naecoms.1	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	sylnbir 333	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-10 2176  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-nf 1805

This theorem is referenced by:  sb9  2551  eujustALT  2600  nfcvf2  2952  axpowndlem2  10567  axsepg2  35440  axsepg4  35443  axnulg  35445  axpowg2  35447  axpowg3  35448  axtcond  36843  mh-setindnd  36902  wl-sbcom2d  38069  wl-mo2df  38078  wl-eudf  38080