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Metamath Proof Explorer

Theorem ax6ev

Description: At least one individual exists. Weaker version of ax6e . When possible, use of this theorem rather than ax6e is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017)

		Ref	Expression
	Assertion	ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦

Proof

Step	Hyp	Ref	Expression
1		ax6v	⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦
2		df-ex	⊢ ( ∃ 𝑥 𝑥 = 𝑦 ↔ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 )
3	1 2	mpbir	⊢ ∃ 𝑥 𝑥 = 𝑦