Theorem axc11v

Description: Version of axc11 with a disjoint variable condition on x and y , which is provable, on top of { ax-1 -- ax-7 }, from ax12v (contrary to axc11 which seems to require the full ax-12 and ax-13 ). (Contributed by NM, 16-May-2008) (Revised by BJ, 6-Jul-2021) (Proof shortened by Wolf Lammen, 11-Oct-2021)

	Ref	Expression
Assertion	axc11v	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		axc16g	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑦 𝜑 ) )
2	1	spsd	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) )