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Theorem bj-nnfbi 34119
Description: If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other. Compare nfbiit 1852. From this and bj-nnfim 34137 and bj-nnfnt 34131, one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 34132) in order not to require sp 2184 (modal T). (Contributed by BJ, 27-Aug-2023.)
Assertion
Ref Expression
bj-nnfbi (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))

Proof of Theorem bj-nnfbi
StepHypRef Expression
1 bj-hbyfrbi 34024 . . 3 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((∃𝑥𝜑𝜑) ↔ (∃𝑥𝜓𝜓)))
2 bj-hbxfrbi 34023 . . 3 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
31, 2anbi12d 633 . 2 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥𝜓𝜓) ∧ (𝜓 → ∀𝑥𝜓))))
4 df-bj-nnf 34118 . 2 (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
5 df-bj-nnf 34118 . 2 (Ⅎ'𝑥𝜓 ↔ ((∃𝑥𝜓𝜓) ∧ (𝜓 → ∀𝑥𝜓)))
63, 4, 53bitr4g 317 1 (((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  Ⅎ'wnnf 34117
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 34118
This theorem is referenced by:  bj-nnfbd  34120  bj-nnfbii  34121
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