Theorem expandrexn 41909

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

Hypothesis

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

Assertion

Ref	Expression
expandrexn	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Proof of Theorem expandrexn

Step	Hyp	Ref	Expression
1		expandrexn.1	. . 3 ⊢ (𝜑 ↔ ¬ 𝜓)
2	1	rexbii 3181	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝜓)
3		df-rex 3070	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜓))
4		exanali 1862	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5	2, 3, 4	3bitri 297	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-rex 3070

This theorem is referenced by:  expandrex  41910  ismnuprim  41912