Theorem ax13w 2135

Description: Weak version (principal instance) of ax-13 2375. (Because 𝑦 and 𝑧 don't need to be distinct, this actually bundles the principal instance and the degenerate instance (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥𝑦 = 𝑦)).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 2128, ax11w 2129, and ax12w 2132. (Contributed by NM, 10-Apr-2017.)

Assertion

Ref	Expression
ax13w	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

Proof of Theorem ax13w

Step	Hyp	Ref	Expression
1		ax5d 1910	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-5 1909

This theorem is referenced by: (None)