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Metamath Proof Explorer

Theorem axc16ALT

Description: Alternate proof of axc16 , shorter but requiring ax-10 , ax-11 , ax-13 and using df-nf and df-sb . (Contributed by NM, 17-May-2008) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	axc16ALT	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbequ12	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 ) )
2		ax-5	⊢ ( 𝜑 → ∀ 𝑧 𝜑 )
3	2	hbsb3	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑧 / 𝑥 ] 𝜑 )
4	1 3	axc16i	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) )