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Theorem bj-nnfim1 34255
 Description: A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.)
Assertion
Ref Expression
bj-nnfim1 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))

Proof of Theorem bj-nnfim1
StepHypRef Expression
1 bj-nnfe 34244 . 2 (Ⅎ'𝑥𝜑 → (∃𝑥𝜑𝜑))
2 bj-nnfa 34242 . 2 (Ⅎ'𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3 imim12 105 . . 3 ((∃𝑥𝜑𝜑) → ((𝜓 → ∀𝑥𝜓) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))))
43imp 410 . 2 (((∃𝑥𝜑𝜑) ∧ (𝜓 → ∀𝑥𝜓)) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
51, 2, 4syl2an 598 1 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎ'wnnf 34237 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 210  df-an 400  df-bj-nnf 34238 This theorem is referenced by:  bj-nnfim  34257
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