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Theorem expandrexn 41407
 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 3175 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 ¬ 𝜓)
3 df-rex 3076 . 2 (∃𝑥𝐴 ¬ 𝜓 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝜓))
4 exanali 1860 . 2 (∃𝑥(𝑥𝐴 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝑥𝐴𝜓))
52, 3, 43bitri 300 1 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-rex 3076 This theorem is referenced by:  expandrex  41408  ismnuprim  41410
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