Metamath Proof Explorer


Theorem axc11rv

Description: Version of axc11r 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 , and to axc11r which seems to require the full ax-12 ). (Contributed by BJ, 6-Jul-2021) (Proof shortened by Wolf Lammen, 11-Oct-2021)

Ref Expression
Assertion axc11rv ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) )

Proof

Step Hyp Ref Expression
1 axc16 ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
2 1 spsd ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) )