Theorem expandexn 41796

Description: Expand an existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypothesis

Ref	Expression
expandexn.1	⊢ (𝜑 ↔ ¬ 𝜓)

Assertion

Ref	Expression
expandexn	⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓)

Proof of Theorem expandexn

Step	Hyp	Ref	Expression
1		expandexn.1	. . 3 ⊢ (𝜑 ↔ ¬ 𝜓)
2	1	exbii 1851	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑥 ¬ 𝜓)
3		exnal 1830	. 2 ⊢ (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
4	2, 3	bitri 274	1 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  rr-grothprimbi  41802