Theorem eximal 1815

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 1864. (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 1813	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2	1	imbi1i 352	. 2 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ ∀𝑥 ¬ 𝜑 → 𝜓))
3		con1b 361	. 2 ⊢ ((¬ ∀𝑥 ¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
4	2, 3	bitri 278	1 ⊢ ((∃𝑥𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  ax5e  1945  axc16nf  2302  xfree2  32834  bj-exalims  37281  bj-dfnnf2  37405  bj-nnfnt  37416