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Theorem expandexn 40632
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
StepHypRef Expression
1 expandexn.1 . . 3 (𝜑 ↔ ¬ 𝜓)
21exbii 1848 . 2 (∃𝑥𝜑 ↔ ∃𝑥 ¬ 𝜓)
3 exnal 1827 . 2 (∃𝑥 ¬ 𝜓 ↔ ¬ ∀𝑥𝜓)
42, 3bitri 277 1 (∃𝑥𝜑 ↔ ¬ ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208  wal 1535  wex 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810
This theorem depends on definitions:  df-bi 209  df-ex 1781
This theorem is referenced by:  rr-grothprimbi  40638
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