Theorem eximal 1784

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 non-freeness. See also alimex 1832. (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 1782	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2	1	imbi1i 353	. 2 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ ∀𝑥 ¬ 𝜑 → 𝜓))
3		con1b 362	. 2 ⊢ ((¬ ∀𝑥 ¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
4	2, 3	bitri 278	1 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781

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 210  df-ex 1782

This theorem is referenced by:  ax5e  1913  axc16nf  2261  xfree2  30228  bj-exalims  34080  bj-dfnnf2  34181  bj-nnfnt  34184