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Metamath Proof Explorer

Theorem alimex

Description: An equivalence between an implication with a universally quantified consequent and an implication with an existentially quantified antecedent. An interesting case is when the same formula is substituted for both ph and ps , since then both implications express a type of nonfreeness. See also eximal . (Contributed by BJ, 12-May-2019)

		Ref	Expression
	Assertion	alimex	⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 ¬ 𝜓 → ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		alex	⊢ ( ∀ 𝑥 𝜓 ↔ ¬ ∃ 𝑥 ¬ 𝜓 )
2	1	imbi2i	⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( 𝜑 → ¬ ∃ 𝑥 ¬ 𝜓 ) )
3		con2b	⊢ ( ( 𝜑 → ¬ ∃ 𝑥 ¬ 𝜓 ) ↔ ( ∃ 𝑥 ¬ 𝜓 → ¬ 𝜑 ) )
4	2 3	bitri	⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 ¬ 𝜓 → ¬ 𝜑 ) )