Theorem alimex 1827

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 𝜑 and 𝜓, since then both implications express a type of non-freeness. See also eximal 1779. (Contributed by BJ, 12-May-2019.)

Assertion

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

Proof of Theorem alimex

Step	Hyp	Ref	Expression
1		alex 1822	. . 3 ⊢ (∀𝑥𝜓 ↔ ¬ ∃𝑥 ¬ 𝜓)
2	1	imbi2i 338	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ¬ ∃𝑥 ¬ 𝜓))
3		con2b 362	. 2 ⊢ ((𝜑 → ¬ ∃𝑥 ¬ 𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
4	2, 3	bitri 277	1 ⊢ ((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by:  bj-nnfnt  34064