Theorem ax4sp1 1426

Description: A special case of ax-4 1400 without using ax-4 1400 or ax-17 1419. (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 1425	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2	1	pm2.21i 575	1 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-i9 1423

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by: (None)