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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-wnf1 | Structured version Visualization version GIF version |
Description: When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.) |
Ref | Expression |
---|---|
bj-wnf1 | ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-modal4e 34897 | . . 3 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑) | |
2 | hba1 2290 | . . 3 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓) | |
3 | 1, 2 | imim12i 62 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥∃𝑥𝜑 → ∀𝑥∀𝑥𝜓)) |
4 | 19.38 1841 | . 2 ⊢ ((∃𝑥∃𝑥𝜑 → ∀𝑥∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)) | |
5 | 3, 4 | syl 17 | 1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: bj-wnfanf 34901 bj-wnfenf 34902 bj-wnfnf 34921 |
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