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Theorem bj-nfexd 34429
Description: Variant of nfexd 2344. (Contributed by BJ, 25-Dec-2023.)
Hypotheses
Ref Expression
bj-nfald.1 (𝜑 → ∀𝑦𝜑)
bj-nfald.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
bj-nfexd (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem bj-nfexd
StepHypRef Expression
1 df-ex 1777 . 2 (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2 bj-nfald.1 . . . 4 (𝜑 → ∀𝑦𝜑)
3 bj-nfald.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
43nfnd 1854 . . . 4 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
52, 4bj-nfald 34428 . . 3 (𝜑 → Ⅎ𝑥𝑦 ¬ 𝜓)
65nfnd 1854 . 2 (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
71, 6nfxfrd 1850 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1531  wex 1776  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1777  df-nf 1781
This theorem is referenced by:  copsex2d  34430
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