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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-wnfanf | Structured version Visualization version GIF version | ||
| Description: When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.) |
| Ref | Expression |
|---|---|
| bj-wnfanf | ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-wnf1 36696 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)) | |
| 2 | bj-19.23bit 36670 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓)) | |
| 3 | 1, 2 | sylg 1823 | 1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
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