Theorem ax4sp1 1526

Description: A special case of ax-4 1503 without using ax-4 1503 or ax-17 1519. (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 1525	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2	1	pm2.21i 641	1 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-8 1497  ax-i9 1523

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by: (None)