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Metamath Proof Explorer

Theorem ax13dgen3

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

		Ref	Expression
	Assertion	ax13dgen3	⊢ ( ¬ 𝑥 = 𝑦 → ( 𝑦 = 𝑦 → ∀ 𝑥 𝑦 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		equid	⊢ 𝑦 = 𝑦
2	1	ax-gen	⊢ ∀ 𝑥 𝑦 = 𝑦
3	2	2a1i	⊢ ( ¬ 𝑥 = 𝑦 → ( 𝑦 = 𝑦 → ∀ 𝑥 𝑦 = 𝑦 ) )