Metamath Proof Explorer


Theorem axc5sp1

Description: A special case of ax-c5 without using ax-c5 or ax-5 . (Contributed by NM, 13-Jan-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion axc5sp1 ( ∀ 𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥 )

Proof

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