Theorem expandrex 44289

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

Hypothesis

Ref	Expression
expandrex.1	⊢ (𝜑 ↔ 𝜓)

Assertion

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

Proof of Theorem expandrex

Step	Hyp	Ref	Expression
1		expandrex.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		notnotb 315	. . 3 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
3	1, 2	bitri 275	. 2 ⊢ (𝜑 ↔ ¬ ¬ 𝜓)
4	3	expandrexn 44288	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2108  ∃wrex 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-rex 3070

This theorem is referenced by:  ismnuprim  44291