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Theorem bj-nalnalimiOLD 31799
 Description: An inference for distributing quantifiers over a double implication. The general statement that spimfw 1865 proves. (Contributed by BJ, 12-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-nalnalimiOLD.1 (𝜒 → (𝜑𝜓))
bj-nalnalimiOLD.2 𝜓 → ∀𝑥 ¬ 𝜓)
Assertion
Ref Expression
bj-nalnalimiOLD (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑𝜓))

Proof of Theorem bj-nalnalimiOLD
StepHypRef Expression
1 bj-nalnalimiOLD.1 . . 3 (𝜒 → (𝜑𝜓))
21bj-nalnaleximiOLD 31798 . 2 (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑 → ∃𝑥𝜓))
3 bj-nalnalimiOLD.2 . . 3 𝜓 → ∀𝑥 ¬ 𝜓)
4 eximal 1698 . . 3 ((∃𝑥𝜓𝜓) ↔ (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
53, 4mpbir 220 . 2 (∃𝑥𝜓𝜓)
62, 5syl6 34 1 (¬ ∀𝑥 ¬ 𝜒 → (∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
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