Theorem eximal 1782

Description: An equivalence between an implication with an existentially quantified antecedent and an implication with a universally quantified consequent. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of nonfreeness. See also alimex 1831. (Contributed by BJ, 12-May-2019.)

Assertion

Ref	Expression
eximal	⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Proof of Theorem eximal

Step	Hyp	Ref	Expression
1		df-ex 1780	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2	1	imbi1i 349	. 2 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ ∀𝑥 ¬ 𝜑 → 𝜓))
3		con1b 358	. 2 ⊢ ((¬ ∀𝑥 ¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
4	2, 3	bitri 275	1 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779

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

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  ax5e  1912  axc16nf  2264  xfree2  32431  bj-exalims  36657  bj-dfnnf2  36760  bj-nnfnt  36763