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Metamath Proof Explorer

Theorem alnex

Description: Universal quantification of negation is equivalent to negation of existential quantification. Dual of exnal (but does not depend on ax-4 contrary to it). See also the dual pair df-ex / alex . Theorem 19.7 of Margaris p. 89. (Contributed by NM, 12-Mar-1993)

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

Proof

Step	Hyp	Ref	Expression
1		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
2	1	con2bii	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 )