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Theorem expandrexn 44287
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 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴𝜓))

Proof of Theorem expandrexn
StepHypRef Expression
1 expandrexn.1 . . 3 (𝜑 ↔ ¬ 𝜓)
21rexbii 3092 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 ¬ 𝜓)
3 df-rex 3069 . 2 (∃𝑥𝐴 ¬ 𝜓 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝜓))
4 exanali 1857 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝑥𝐴𝜓))
52, 3, 43bitri 297 1 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535  wex 1776  wcel 2106  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-rex 3069
This theorem is referenced by:  expandrex  44288  ismnuprim  44290
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