This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ax13dgen2

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

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

Proof

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