Theorem ax13dgen2 2137

Description: Degenerate instance of ax-13 2373 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)

Assertion

Ref	Expression
ax13dgen2	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))

Proof of Theorem ax13dgen2

Step	Hyp	Ref	Expression
1		equcomi 2023	. 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
2		pm2.21 123	. 2 ⊢ (¬ 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑥 𝑦 = 𝑥))
3	1, 2	syl5 34	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786

This theorem is referenced by: (None)