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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfim | Structured version Visualization version GIF version |
Description: Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.) |
Ref | Expression |
---|---|
bj-nnfim | ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1881 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | bj-nnfim2 35239 | . . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → 𝜓))) | |
3 | 1, 2 | biimtrid 241 | . 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))) |
4 | bj-nnfim1 35238 | . . 3 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | |
5 | 19.38 1842 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
6 | 4, 5 | syl6 35 | . 2 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))) |
7 | df-bj-nnf 35218 | . 2 ⊢ (Ⅎ'𝑥(𝜑 → 𝜓) ↔ ((∃𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)) ∧ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))) | |
8 | 3, 6, 7 | sylanbrc 584 | 1 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓)) |
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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