Theorem naecoms 2428

Description: A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2371. (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 2426	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
2		naecoms.1	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	sylnbir 331	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2143  ax-12 2179  ax-13 2371

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by:  sb9  2518  eujustALT  2566  nfcvf2  2920  axpowndlem2  10481  axnulg  35094  wl-sbcom2d  37574  wl-mo2df  37583  wl-eudf  37585