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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 ) )