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Theorem bj-nnfim 35240
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 1881 . . 3 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 bj-nnfim2 35239 . . 3 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑𝜓)))
31, 2biimtrid 241 . 2 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → (∃𝑥(𝜑𝜓) → (𝜑𝜓)))
4 bj-nnfim1 35238 . . 3 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
5 19.38 1842 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
64, 5syl6 35 . 2 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑𝜓) → ∀𝑥(𝜑𝜓)))
7 df-bj-nnf 35218 . 2 (Ⅎ'𝑥(𝜑𝜓) ↔ ((∃𝑥(𝜑𝜓) → (𝜑𝜓)) ∧ ((𝜑𝜓) → ∀𝑥(𝜑𝜓))))
83, 6, 7sylanbrc 584 1 ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wex 1782  Ⅎ'wnnf 35217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-bj-nnf 35218
This theorem is referenced by:  bj-nnfimd  35241  bj-nnfbit  35246  bj-nnfbid  35247
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