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Metamath Proof Explorer

Theorem axc16nfALT

Description: Alternate proof of axc16nf , shorter but requiring ax-11 and ax-13 . (Contributed by Mario Carneiro, 7-Oct-2016) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	axc16nfALT	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑧 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfae	⊢ Ⅎ 𝑧 ∀ 𝑥 𝑥 = 𝑦
2		axc16g	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑧 𝜑 ) )
3	1 2	nf5d	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑧 𝜑 )