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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfbi | Structured version Visualization version GIF version |
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 1858. From this and bj-nnfim 34699 and bj-nnfnt 34693, 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 34694) in order not to require sp 2182 (modal T). (Contributed by BJ, 27-Aug-2023.) |
Ref | Expression |
---|---|
bj-nnfbi | ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-hbyfrbi 34583 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((∃𝑥𝜑 → 𝜑) ↔ (∃𝑥𝜓 → 𝜓))) | |
2 | bj-hbxfrbi 34582 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))) | |
3 | 1, 2 | anbi12d 634 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ↔ ((∃𝑥𝜓 → 𝜓) ∧ (𝜓 → ∀𝑥𝜓)))) |
4 | df-bj-nnf 34677 | . 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | |
5 | df-bj-nnf 34677 | . 2 ⊢ (Ⅎ'𝑥𝜓 ↔ ((∃𝑥𝜓 → 𝜓) ∧ (𝜓 → ∀𝑥𝜓))) | |
6 | 3, 4, 5 | 3bitr4g 317 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 ∃wex 1787 Ⅎ'wnnf 34676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-bj-nnf 34677 |
This theorem is referenced by: bj-nnfbd 34679 bj-nnfbii 34680 |
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