Theorem ax4sp1 1579

Description: A special case of ax-4 1556 without using ax-4 1556 or ax-17 1572. (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 1578	. 2 ⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
2	1	pm2.21i 649	1 ⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie2 1540  ax-8 1550  ax-i9 1576

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401

This theorem is referenced by: (None)