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Metamath Proof Explorer

Theorem axc11

Description: Show that ax-c11 can be derived from ax-c11n in the form of axc11n . Normally, axc11 should be used rather than ax-c11 , except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker axc11v when possible. (Contributed by NM, 16-May-2008) (Proof shortened by Wolf Lammen, 21-Apr-2018) (New usage is discouraged.)

		Ref	Expression
	Assertion	axc11	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		axc11r	⊢ ( ∀ 𝑦 𝑦 = 𝑥 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) )
2	1	aecoms	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜑 ) )