Theorem ax4sp1 1466

Description: A special case of ax-4 1440 without using ax-4 1440 or ax-17 1459. (Contributed by NM, 13-Jan-2011.)

Assertion

Ref	Expression
ax4sp1	⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Proof of Theorem ax4sp1

Step	Hyp	Ref	Expression
1		equidqe 1465	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2	1	pm2.21i 607	1 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-8 1435  ax-i9 1463

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by: (None)