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Theorem bj-nnfim 34497
 Description: Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.)
Assertion
Ref Expression
bj-nnfim ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))

Proof of Theorem bj-nnfim
StepHypRef Expression
1 19.35 1878 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 bj-nnfim2 34496 . . 3 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑𝜓)))
31, 2syl5bi 245 . 2 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
4 bj-nnfim1 34495 . . 3 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5 19.38 1840 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
64, 5syl6 35 . 2 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → ∀𝑥(𝜑𝜓)))
7 df-bj-nnf 34478 . 2 (Ⅎ'𝑥(𝜑𝜓) ↔ ((∃𝑥(𝜑𝜓) → (𝜑𝜓)) ∧ ((𝜑𝜓) → ∀𝑥(𝜑𝜓))))
83, 6, 7sylanbrc 586 1 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎ'wnnf 34477 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-bj-nnf 34478 This theorem is referenced by:  bj-nnfimd  34498  bj-nnfbit  34503  bj-nnfbid  34504
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