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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfim1 | Structured version Visualization version GIF version |
Description: A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.) |
Ref | Expression |
---|---|
bj-nnfim1 | ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnfe 34650 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | |
2 | bj-nnfa 34647 | . 2 ⊢ (Ⅎ'𝑥𝜓 → (𝜓 → ∀𝑥𝜓)) | |
3 | imim12 105 | . . 3 ⊢ ((∃𝑥𝜑 → 𝜑) → ((𝜓 → ∀𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))) | |
4 | 3 | imp 410 | . 2 ⊢ (((∃𝑥𝜑 → 𝜑) ∧ (𝜓 → ∀𝑥𝜓)) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
5 | 1, 2, 4 | syl2an 599 | 1 ⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 ∃wex 1787 Ⅎ'wnnf 34642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-bj-nnf 34643 |
This theorem is referenced by: bj-nnfim 34665 |
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