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Metamath Proof Explorer

Theorem alex

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex . See also the dual pair alnex / exnal . Theorem 19.6 of Margaris p. 89. (Contributed by NM, 12-Mar-1993)

		Ref	Expression
	Assertion	alex	⊢ ( ∀ 𝑥 𝜑 ↔ ¬ ∃ 𝑥 ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		notnotb	⊢ ( 𝜑 ↔ ¬ ¬ 𝜑 )
2	1	albii	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 ¬ ¬ 𝜑 )
3		alnex	⊢ ( ∀ 𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃ 𝑥 ¬ 𝜑 )
4	2 3	bitri	⊢ ( ∀ 𝑥 𝜑 ↔ ¬ ∃ 𝑥 ¬ 𝜑 )