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Metamath Proof Explorer

Theorem axc16b

Description: This theorem shows that Axiom ax-c16 is redundant in the presence of Theorem dtruALT2 , which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006)

		Ref	Expression
	Assertion	axc16b	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		dtruALT2	⊢ ¬ ∀ 𝑥 𝑥 = 𝑦
2	1	pm2.21i	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 𝜑 ) )