Metamath Proof Explorer


Theorem bj-axc11v

Description: Version of axc11 with a disjoint variable condition, which does not require ax-13 nor ax-10 . Remark: the following theorems ( hbae , nfae , hbnae , nfnae , hbnaes ) would need to be totally unbundled to be proved without ax-13 , hence would be simple consequences of ax-5 or nfv . (Contributed by BJ, 31-May-2019) (Proof modification is discouraged.)

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

Proof

Step Hyp Ref Expression
1 axc11rv ( ∀ 𝑦 𝑦 = 𝑥 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) )
2 1 bj-aecomsv ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) )