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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 ) )